Solving One-Step and Multi-Step Equations
| Site: | Young Education |
| Course: | Introduction to Algebra |
| Book: | Solving One-Step and Multi-Step Equations |
| Printed by: | Guest user |
| Date: | Monday, 9 November 2026, 4:39 PM |
1. Understanding Equations
Learning outcomes
- I can explain the meaning of an equation.
- I can identify the parts of an equation.
- I can describe equality as a balance between two sides.
- I can determine whether a value is a solution to an equation.
- I can verify solutions by substitution.
Understanding Equations
An equation is a mathematical statement showing that two expressions have the same value.
For example:
x + 4 = 10
The equals sign tells us that:
x + 4 has the same value as 10
An equation is like a balanced scale. The expressions on the left and right may look different, but their values must be equal.
What Is an Equation?
An equation contains an:
equals sign (=)
Examples:
5 + 3 = 8
x + 7 = 12
2x = 14
3x − 4 = 11
y = 2x + 5
Each equation states that the expression on the left has the same value as the expression on the right.
Expressions and Equations
An expression contains numbers, variables, and operations but does not state that two quantities are equal.
Examples of expressions:
3x + 5
7 − y
2a + 4b
An equation contains an equals sign:
3x + 5 = 20
So:
3x + 5 → expression
3x + 5 = 20 → equation
The Two Sides of an Equation
Every equation has two sides.
Consider:
2x + 3 = 11
The left-hand side (LHS) is:
2x + 3
The right-hand side (RHS) is:
11
The equals sign states:
LHS = RHS
Parts of an Equation
Consider:
3x + 7 = 22
We can identify several important parts.
Variable: x
Coefficient: 3
Constant: 7 and 22
Terms on the left: 3x and 7
Equals sign: =
These parts help us describe and work with equations.
Variables
A variable is a letter or symbol representing a number.
Examples include:
x, y, a, n
In:
x + 5 = 12
the variable is:
x
The goal may be to determine which value of x makes the equation true.
Coefficients
A coefficient is a number multiplying a variable.
In:
4x + 3 = 19
the coefficient of x is:
4
because:
4x = 4 × x
In:
−2y = 10
the coefficient of y is:
−2
Constants
A constant is a number whose value does not change within the expression.
In:
5x + 8 = 23
the numbers:
8 and 23
are constants.
The number 5 is specifically the coefficient of x.
Terms
A term is a number, variable, or product of numbers and variables separated by addition or subtraction.
For example:
4x + 7 − 2y
contains the terms:
4x
7
−2y
Recognizing terms becomes important when simplifying and solving equations.
Operations
Equations can contain mathematical operations such as:
- addition (+)
- subtraction (−)
- multiplication (×)
- division (÷)
- powers
For example:
3x + 5 = 20
contains multiplication and addition.
The Equals Sign
The equals sign is one of the most important symbols in mathematics.
= means "has the same value as"
It does not mean:
"the answer comes next."
For example:
4 + 5 = 6 + 3
is true because:
9 = 9
Both sides have the same value.
Equality as Balance
A useful model for an equation is a:
balance scale
Imagine:
x + 3 = 8
On one side of a balanced scale are:
- one unknown box
- three unit masses
On the other side are:
- eight unit masses
Because the scale balances, both sides must represent the same total amount.
Maintaining Equality
Suppose:
x + 3 = 8
If we remove 3 from the left side, we must also remove 3 from the right side.
x + 3 − 3 = 8 − 3
Therefore:
x = 5
The equation remains balanced because we performed the same operation on both sides.
This gives one of the most important rules of algebra:
Whatever you do to one side of an equation, you must do to the other side.
Addition Property of Equality
If:
a = b
then:
a + c = b + c
Example:
x − 4 = 7
Add 4 to both sides:
x − 4 + 4 = 7 + 4
Therefore:
x = 11
Subtraction Property of Equality
If:
a = b
then:
a − c = b − c
Example:
x + 6 = 15
Subtract 6 from both sides:
x + 6 − 6 = 15 − 6
Therefore:
x = 9
Multiplication Property of Equality
If:
a = b
then:
ac = bc
Example:
x/4 = 3
Multiply both sides by 4:
4(x/4) = 4(3)
Therefore:
x = 12
Division Property of Equality
If:
a = b
then, for c ≠ 0:
a/c = b/c
Example:
5x = 30
Divide both sides by 5:
5x/5 = 30/5
Therefore:
x = 6
What Is a Solution?
A solution to an equation is a value that makes the equation true.
Consider:
x + 4 = 9
Try:
x = 5
Substitute 5 for x:
5 + 4 = 9
9 = 9
The statement is true.
Therefore:
x = 5 is a solution
Testing a Value
Consider:
2x + 1 = 11
Is:
x = 5
a solution?
Substitute:
2(5) + 1 = 11
10 + 1 = 11
11 = 11
Yes.
Therefore:
x = 5 is a solution
Testing an Incorrect Value
Consider again:
2x + 1 = 11
Test:
x = 4
Substitute:
2(4) + 1 = 11
8 + 1 = 11
9 = 11
This statement is false.
Therefore:
x = 4 is not a solution
Substitution
Substitution means replacing a variable with a given value.
Suppose:
3x − 2 = 13
and we want to test:
x = 5
Replace every x with 5:
3(5) − 2 = 13
Then evaluate:
15 − 2 = 13
13 = 13
Therefore the solution is verified.
Why Substitution Is Important
Substitution allows us to check our work.
Suppose we solve an equation and obtain:
x = 7
We should substitute 7 into the original equation.
If both sides become equal, our solution is verified.
This is especially useful for catching:
- arithmetic mistakes
- sign errors
- incorrect inverse operations
Worked Example: Checking a Solution
Equation:
4x + 3 = 27
Suppose we think:
x = 6
Substitute:
4(6) + 3 = 27
24 + 3 = 27
27 = 27
The equation is true.
Therefore:
x = 6 is correct.
Worked Example: Is It a Solution?
Equation:
5x − 4 = 21
Test:
x = 4
Substitute:
5(4) − 4 = 21
20 − 4 = 21
16 = 21
False.
Therefore:
x = 4 is not a solution.
Now test:
x = 5
5(5) − 4 = 21
25 − 4 = 21
21 = 21
True.
Therefore:
x = 5 is a solution.
Worked Example: Negative Numbers
Equation:
2x + 5 = −3
Test:
x = −4
Substitute:
2(−4) + 5 = −3
−8 + 5 = −3
−3 = −3
True.
Therefore:
x = −4 is a solution.
Worked Example: Variables on Both Sides
Consider:
3x + 2 = x + 10
Test:
x = 4
Left-hand side:
3(4) + 2 = 14
Right-hand side:
4 + 10 = 14
Therefore:
14 = 14
So:
x = 4 is a solution.
Equations Can Look Different but Be Equivalent
Consider:
x + 3 = 10
Subtract 3 from both sides:
x = 7
These equations look different, but they have the same solution.
They are:
equivalent equations
Other equivalent equations include:
2x = 14
3x = 21
Each has:
x = 7
Equations as Relationships
Not every equation asks us to find one unknown number.
For example:
d = vt
describes a relationship between:
- distance d
- velocity v
- time t
Similarly:
A = lw
relates:
- area A
- length l
- width w
Equations are therefore useful for representing relationships between quantities.
Equations in Science
Equations are extremely important in science.
Examples include:
v = d/t
F = ma
ρ = m/V
These equations describe relationships between physical quantities.
For example:
F = ma
means that net force equals mass multiplied by acceleration.
Equations in Everyday Life
Suppose a taxi charges:
$4 starting fee + $2 per kilometer
If x is the distance traveled and C is the total cost:
C = 4 + 2x
For a 5 km journey:
C = 4 + 2(5)
C = 14
Equations allow us to represent and predict real situations.
Equations from Words
Mathematical statements can be translated into equations.
Example
"A number plus 7 equals 15."
Let the number be x.
x + 7 = 15
Example
"Three times a number equals 24."
3x = 24
Example
"Five less than twice a number equals 13."
2x − 5 = 13
Learning to translate between words and equations is an important algebra skill.
True and False Equations
Some equations contain no variables.
For example:
7 + 5 = 12
This is:
true
But:
7 + 5 = 15
is:
false
An equation containing a variable becomes true only for certain values—or, in some cases, for every value or no value.
Equations True for Every Value
Consider:
2(x + 3) = 2x + 6
Expand the left side:
2x + 6 = 2x + 6
This is true for every value of x.
Such an equation is called an:
identity
Equations with No Solution
Consider:
x + 4 = x + 7
Subtract x from both sides:
4 = 7
This is impossible.
Therefore the equation has:
no solution
One Solution, No Solution, or Many Solutions
Equations can have different numbers of solutions.
One solution
x + 3 = 8
Solution:
x = 5
No solution
x + 3 = x + 7
This would require:
3 = 7
Impossible.
Infinitely many solutions
2(x + 3) = 2x + 6
Both sides are equivalent for every x.
A Systematic Way to Verify a Solution
When checking a proposed solution:
Step 1: Write the original equation.
Step 2: Substitute the proposed value for every occurrence of the variable.
Step 3: Evaluate the left-hand side.
Step 4: Evaluate the right-hand side.
Step 5: Compare the results.
If:
LHS = RHS
the value is a solution.
If:
LHS ≠ RHS
the value is not a solution.
Worked Example: Full Verification
Equation:
3x − 7 = 2x + 2
Suppose:
x = 9
Left-hand side
3(9) − 7
= 27 − 7
= 20
Right-hand side
2(9) + 2
= 18 + 2
= 20
Therefore:
LHS = RHS
and:
x = 9 is verified.
Worked Example: Detecting an Error
A student solves:
4x + 5 = 17
and claims:
x = 4
Check using substitution:
4(4) + 5 = 17
16 + 5 = 17
21 = 17
False.
Therefore the proposed solution is incorrect.
Solving correctly:
4x + 5 = 17
Subtract 5:
4x = 12
Divide by 4:
x = 3
Check:
4(3) + 5 = 17
12 + 5 = 17
17 = 17
Correct.
Why Balance Matters
The balance idea is the foundation of solving equations.
If:
LHS = RHS
and we change only one side, the equality may become false.
For example:
8 = 8
Subtract 3 only from the left:
5 ≠ 8
But subtract 3 from both sides:
5 = 5
The balance is maintained.
Common Mistakes
Mistake: Thinking = means "the answer is next."
The equals sign means both sides have the same value.
Mistake: Confusing an expression with an equation.
An equation contains an equals sign; an expression does not.
Mistake: Changing only one side of an equation.
When solving using equality-preserving operations, perform the corresponding operation on both sides.
Mistake: Confusing coefficients and constants.
In 5x + 3 = 18, 5 is the coefficient of x, while 3 and 18 are constants.
Mistake: Assuming a proposed value must be a solution.
Always test it by substitution.
Mistake: Substituting into only one occurrence of a variable.
Replace every occurrence of the variable.
Mistake: Checking against a modified equation instead of the original equation.
The strongest verification is substitution into the original equation.
Mistake: Forgetting parentheses around negative substitutions.
If x = −3, write 2(−3), not an ambiguous expression.
Key Terms
Equation — mathematical statement showing that two expressions have equal values.
Equals sign (=) — symbol indicating equality between two quantities or expressions.
Expression — mathematical combination of numbers, variables, and operations without an equality statement.
Left-hand side (LHS) — expression to the left of the equals sign.
Right-hand side (RHS) — expression to the right of the equals sign.
Variable — symbol representing a number or quantity.
Coefficient — numerical factor multiplying a variable.
Constant — number whose value does not change within an expression.
Term — part of an expression separated by addition or subtraction.
Solution — value that makes an equation true.
Substitution — replacing a variable with a particular value.
Verification — checking that a proposed solution makes the original equation true.
Equivalent equations — equations with the same solution set.
Identity — equation true for every permissible value of its variable.
Inverse operations — operations that undo each other, such as addition and subtraction or multiplication and division.
Key Takeaways
- An equation states that two expressions have equal values.
- The equals sign means "has the same value as."
- An equation has a left-hand side and a right-hand side.
- Variables represent unknown or changing quantities.
- Coefficients multiply variables.
- Constants have fixed numerical values.
- Terms are separated by addition or subtraction.
- Equality can be understood using the model of a balanced scale.
- Performing the same valid operation on both sides maintains equality.
- A solution is a value that makes an equation true.
- Substitution is used to test possible solutions.
- If substitution produces LHS = RHS, the proposed value is a solution.
- If substitution produces LHS ≠ RHS, the proposed value is not a solution.
- Solutions should be verified using the original equation.
- Equivalent equations have the same solution set.
- Some equations have one solution.
- Some equations have no solution.
- Some equations are true for every permissible value.
- Equations can represent relationships between quantities.
- Equations are fundamental tools in mathematics, science, engineering, economics, and everyday problem solving.
Check Your Understanding
- What is an equation?
- What does the equals sign mean?
- Explain the difference between an expression and an equation.
- Identify the variable in 5x + 7 = 32.
- Identify the coefficient of x in 5x + 7 = 32.
- Identify the constants in 5x + 7 = 32.
- What is the left-hand side of 3y − 4 = 17?
- What is the right-hand side?
- Explain why an equation can be compared with a balanced scale.
- If 7 is added to one side of an equation, what must be done to the other side to preserve equality?
- Is x = 4 a solution to 2x + 3 = 11? Verify by substitution.
- Is x = 5 a solution to 3x − 2 = 10? Explain.
- Verify whether x = −2 solves 4x + 3 = −5.
- Verify whether y = 6 solves 2y − 5 = y + 1.
- Explain what substitution means.
- Why is it useful to substitute a solution back into the original equation?
- Give an example of an equation with no solution.
- Give an example of an equation that is true for every value of x.
- Write an equation for: "Three times a number plus 4 equals 19."
- Explain in your own words why maintaining equality is the central idea behind solving equations.
2. Solving One-Step Equations
Learning outcomes
- I can solve equations involving addition and subtraction.
- I can solve equations involving multiplication and division.
- I can use inverse operations to isolate variables.
- I can check my solutions for accuracy.
- I can explain the steps used to solve an equation.
Solving One-Step Equations
A one-step equation is an equation that can be solved using one inverse operation.
For example:
x + 5 = 12
can be solved by subtracting 5 from both sides:
x = 7
The goal when solving an equation is to isolate the variable while keeping both sides equal.
The Main Goal: Isolate the Variable
To isolate the variable means to get the variable by itself on one side of the equation.
For example:
x + 8 = 15
We want:
x = ?
Since 8 is being added to x, use the opposite operation:
subtract 8
x + 8 − 8 = 15 − 8
Therefore:
x = 7
Equations Must Stay Balanced
An equation represents two equal quantities.
If:
x + 4 = 10
then both sides have the same value.
Whatever operation we perform on one side must also be performed on the other side.
So:
x + 4 − 4 = 10 − 4
which gives:
x = 6
This preserves the equality.
Inverse Operations
Inverse operations are operations that undo each other.
| Operation | Inverse Operation |
|---|---|
| Addition | Subtraction |
| Subtraction | Addition |
| Multiplication | Division |
| Division | Multiplication |
For example:
+ 7 is undone by − 7
− 5 is undone by + 5
× 4 is undone by ÷ 4
÷ 3 is undone by × 3
Inverse operations are the key to solving one-step equations.
Addition Equations
Consider:
x + 6 = 14
We need to remove the +6.
Use the inverse operation:
−6
Subtract 6 from both sides:
x + 6 − 6 = 14 − 6
Therefore:
x = 8
Checking the Solution
Always check by substituting the solution into the original equation.
Original:
x + 6 = 14
Solution:
x = 8
Substitute:
8 + 6 = 14
14 = 14
The equation is true.
Therefore:
x = 8 is correct.
Worked Example: Addition
Solve:
x + 13 = 25
Subtract 13 from both sides:
x + 13 − 13 = 25 − 13
x = 12
Check:
12 + 13 = 25
25 = 25
Therefore:
x = 12
Addition with Negative Numbers
Solve:
x + 7 = −3
Subtract 7:
x + 7 − 7 = −3 − 7
x = −10
Check:
−10 + 7 = −3
−3 = −3
Correct.
Subtraction Equations
Consider:
x − 5 = 9
The inverse of subtracting 5 is:
adding 5
Add 5 to both sides:
x − 5 + 5 = 9 + 5
Therefore:
x = 14
Worked Example: Subtraction
Solve:
x − 12 = 20
Add 12 to both sides:
x − 12 + 12 = 20 + 12
x = 32
Check:
32 − 12 = 20
20 = 20
Therefore:
x = 32
Subtraction with Negative Answers
Solve:
x − 4 = −9
Add 4 to both sides:
x − 4 + 4 = −9 + 4
x = −5
Check:
−5 − 4 = −9
−9 = −9
Correct.
Multiplication Equations
Consider:
3x = 18
This means:
3 × x = 18
To undo multiplication by 3, divide by 3.
3x/3 = 18/3
Therefore:
x = 6
Worked Example: Multiplication
Solve:
7x = 42
Divide both sides by 7:
7x/7 = 42/7
x = 6
Check:
7(6) = 42
42 = 42
Therefore:
x = 6
Multiplication with Negative Numbers
Solve:
−4x = 28
Divide both sides by −4:
−4x/−4 = 28/−4
x = −7
Check:
−4(−7) = 28
28 = 28
Correct.
When the Coefficient Is Negative
Consider:
−5x = −35
Divide both sides by −5:
x = (−35)/(−5)
x = 7
Remember:
negative ÷ negative = positive
Check:
−5(7) = −35
Correct.
Division Equations
Consider:
x/4 = 7
The variable is being divided by 4.
The inverse operation is:
multiply by 4
Multiply both sides by 4:
4(x/4) = 4(7)
Therefore:
x = 28
Worked Example: Division
Solve:
x/6 = 5
Multiply both sides by 6:
6(x/6) = 6(5)
x = 30
Check:
30/6 = 5
5 = 5
Therefore:
x = 30
Division with Negative Numbers
Solve:
x/−3 = 8
Multiply both sides by −3:
x = 8(−3)
x = −24
Check:
−24/−3 = 8
8 = 8
Correct.
Four Basic Types
Most introductory one-step equations fall into four categories.
Addition
x + a = b
Subtract a:
x = b − a
Subtraction
x − a = b
Add a:
x = b + a
Multiplication
ax = b
Divide by a:
x = b/a
Division
x/a = b
Multiply by a:
x = ab
Choosing the Correct Inverse Operation
Ask:
What operation is being performed on the variable?
Then use the opposite operation.
Example:
x + 9 = 16
Operation on x:
+9
Inverse:
−9
Example:
x − 7 = 18
Operation:
−7
Inverse:
+7
Example:
8x = 56
Operation:
×8
Inverse:
÷8
Example:
x/5 = 9
Operation:
÷5
Inverse:
×5
A Simple Solving Strategy
When solving a one-step equation:
Identify → Undo → Balance → Simplify → Check
Identify
Determine the operation acting on the variable.
Undo
Choose the inverse operation.
Balance
Perform the same operation on both sides.
Simplify
Find the value of the variable.
Check
Substitute your answer into the original equation.
Worked Example: Explain the Steps
Solve:
x + 17 = 31
Step 1: Identify the operation
17 is being added to x.
Step 2: Choose the inverse
Subtract 17.
Step 3: Apply it to both sides
x + 17 − 17 = 31 − 17
Step 4: Simplify
x = 14
Step 5: Check
14 + 17 = 31
31 = 31
Therefore:
x = 14
Worked Example: Multiplication
Solve:
9x = 63
The variable is multiplied by 9.
Use the inverse operation:
divide by 9
9x/9 = 63/9
x = 7
Check:
9(7) = 63
63 = 63
Therefore:
x = 7
Worked Example: Decimal
Solve:
x + 2.5 = 7.8
Subtract 2.5:
x = 7.8 − 2.5
x = 5.3
Check:
5.3 + 2.5 = 7.8
Correct.
Worked Example: Decimal Multiplication
Solve:
0.5x = 8
Divide by 0.5:
x = 8/0.5
x = 16
Check:
0.5(16) = 8
Correct.
Equations with Fractions
The same principles work with fractions.
Solve:
x + 1/4 = 3/4
Subtract 1/4:
x = 3/4 − 1/4
x = 2/4
x = 1/2
Check:
1/2 + 1/4 = 3/4
Correct.
Fraction as a Coefficient
Solve:
x/3 = 4
This can also be written:
(1/3)x = 4
Multiply both sides by 3:
x = 12
Multiplying by a Fraction
Solve:
(1/2)x = 6
The variable is multiplied by 1/2.
We can divide by 1/2:
x = 6 ÷ 1/2
Dividing by 1/2 is the same as multiplying by 2:
x = 12
Using Reciprocals
Suppose:
(2/3)x = 8
The reciprocal of 2/3 is:
3/2
Multiply both sides by 3/2:
x = 8 × 3/2
x = 12
Check:
(2/3)(12) = 8
Correct.
Variables Do Not Have to Be x
Any letter can represent a variable.
Solve:
y + 4 = 15
y = 11
Solve:
5a = 40
a = 8
Solve:
m/7 = 3
m = 21
The method does not change.
Checking Addition Equations
Suppose:
n + 8 = 21
Solution:
n = 13
Substitute:
13 + 8 = 21
21 = 21
Therefore the solution is verified.
Checking Multiplication Equations
Suppose:
6p = 54
Solution:
p = 9
Substitute:
6(9) = 54
54 = 54
Verified.
Detecting an Incorrect Solution
Suppose a student solves:
x − 8 = 15
and writes:
x = 7
Check:
7 − 8 = 15
−1 = 15
False.
Therefore:
x = 7 is incorrect
The correct inverse operation is to add 8:
x = 23
Check:
23 − 8 = 15
Correct.
Why We Use the Same Operation on Both Sides
Consider:
5x = 20
If we divide only the left side by 5:
x = 20
we have changed the equation incorrectly.
Instead:
5x/5 = 20/5
which gives:
x = 4
The equation remains balanced.
One-Step Equations in Word Problems
Equations can represent everyday situations.
Example
A student already has $12 and wants a total of $30.
Let x represent the additional money needed.
x + 12 = 30
Subtract 12:
x = 18
The student needs:
$18 more
Worked Word Problem: Temperature
The temperature increased by 7°C and became 19°C.
Let T represent the original temperature.
T + 7 = 19
Subtract 7:
T = 12
The original temperature was:
12°C
Worked Word Problem: Equal Groups
Five identical notebooks cost $30 altogether.
Let n be the price of one notebook.
5n = 30
Divide by 5:
n = 6
Each notebook costs:
$6
Worked Word Problem: Sharing
A total amount x is divided equally among 4 students.
Each student receives 15 items.
Equation:
x/4 = 15
Multiply by 4:
x = 60
There were:
60 items
in total.
Equations in Science
One-step equation skills are useful when rearranging scientific equations.
Suppose:
F = ma
and:
F = 40 N
m = 5 kg
Then:
40 = 5a
Divide by 5:
a = 8 m/s²
The same algebraic principle is being used.
Another Science Example
Density is:
ρ = m/V
Suppose:
ρ = 4 g/cm³
and:
V = 5 cm³
Then:
4 = m/5
Multiply both sides by 5:
m = 20 g
Again, this is a one-step equation.
Addition and Subtraction Are Inverses
Think of moving along a number line.
If you move:
+5
you can return to your starting position using:
−5
Likewise:
−8
can be undone by:
+8
This is why addition and subtraction are inverse operations.
Multiplication and Division Are Inverses
Suppose a number is multiplied by 4.
To undo that operation:
divide by 4
For example:
7 × 4 = 28
and:
28 ÷ 4 = 7
The operations undo each other.
This is why multiplication equations are solved using division.
Explaining Your Reasoning
A strong mathematical explanation should say:
- what operation is acting on the variable
- which inverse operation is needed
- that the operation is applied to both sides
- what the solution is
- how the solution was checked
For example:
In x − 6 = 13, 6 is being subtracted from x. I add 6 to both sides because addition is the inverse of subtraction. This gives x = 19. Substituting 19 into the original equation gives 19 − 6 = 13, so the solution is correct.
Common Mistakes
Mistake: Using the same operation instead of the inverse.
For:
x + 7 = 12
do not add 7.
Subtract 7.
Mistake: Changing only one side.
Incorrect:
x + 5 = 11
x = 11 − 5
without understanding the balancing step.
Better reasoning:
x + 5 − 5 = 11 − 5
Mistake: Dividing when the variable is already divided.
For:
x/4 = 6
multiply by 4.
Do not divide by 4 again.
Mistake: Ignoring negative signs.
For:
−3x = 12
divide by:
−3
not 3.
Therefore:
x = −4
Mistake: Forgetting to check the solution.
Substitution can quickly reveal an arithmetic or sign error.
Mistake: Checking using a different equation.
Always substitute into the:
original equation
whenever possible.
Key Terms
One-step equation — equation that can be solved using one inverse operation.
Variable — letter or symbol representing an unknown or changing value.
Solution — value that makes an equation true.
Inverse operations — operations that undo one another.
Addition — operation undone by subtraction.
Subtraction — operation undone by addition.
Multiplication — operation undone by division.
Division — operation undone by multiplication.
Isolate — get a variable alone on one side of an equation.
Equality — condition in which two quantities have the same value.
Coefficient — number multiplying a variable.
Substitution — replacing a variable with a specific value.
Verification — checking that a proposed solution makes the original equation true.
Reciprocal — number obtained by reversing the numerator and denominator of a nonzero fraction.
Key Takeaways
- A one-step equation requires one inverse operation to solve.
- The goal is to isolate the variable.
- Addition and subtraction are inverse operations.
- Multiplication and division are inverse operations.
- Whatever operation is performed on one side must also be performed on the other side.
- For x + a = b, subtract a from both sides.
- For x − a = b, add a to both sides.
- For ax = b, divide both sides by a.
- For x/a = b, multiply both sides by a.
- The same methods work with positive numbers, negative numbers, decimals, and fractions.
- A fractional coefficient can often be removed using its reciprocal.
- Variables can be represented by any letter.
- Substitution is used to check a solution.
- A correct solution makes the left-hand side equal the right-hand side.
- Checking the original equation can reveal arithmetic and sign errors.
- One-step equations can model real-world situations.
- The same algebraic skills are used when working with equations in science.
- A good solution should show both the mathematical steps and the reasoning behind them.
Check Your Understanding
- What is a one-step equation?
- What does it mean to isolate a variable?
- What is the inverse of addition?
- What is the inverse of multiplication?
- Solve: x + 7 = 19
- Solve: x − 9 = 14
- Solve: 5x = 35
- Solve: x/6 = 8
- Solve: x + 12 = −4
- Solve: x − 7 = −15
- Solve: −4x = 28
- Solve: x/−5 = 6
- Solve: 0.5x = 9
- Solve: (2/3)x = 10
- Verify whether x = 8 is a solution to x + 5 = 13.
- Verify whether x = −3 is a solution to 4x = −12.
- A number decreased by 9 equals 21. Write and solve an equation.
- Six identical items cost $42. Write and solve an equation to find the cost of one item.
- A student solves x/5 = 7 by dividing both sides by 5. Explain the mistake and give the correct solution.
- Explain why inverse operations and maintaining equality are the two central ideas used when solving one-step equations.
3. Solving Multi-Step Equations
Learning outcomes
- I can solve equations involving multiple operations.
- I can apply inverse operations in the correct order.
- I can isolate variables systematically.
- I can verify solutions by substitution.
- I can explain my reasoning when solving equations.
Solving Multi-Step Equations
A multi-step equation requires more than one operation to isolate the variable.
For example:
3x + 5 = 20
The variable x is first multiplied by 3 and then 5 is added.
To solve the equation, we undo these operations in the reverse order:
3x + 5 = 20
Subtract 5:
3x = 15
Divide by 3:
x = 5
Multi-step equations build directly on the same balance and inverse-operation ideas used for one-step equations.
The Goal: Isolate the Variable
To isolate the variable means to get the variable by itself on one side of the equation.
For:
4x − 7 = 21
we want to reach:
x = ?
There are two operations affecting x:
×4
and:
−7
We undo them systematically.
Add 7:
4x = 28
Divide by 4:
x = 7
Keep the Equation Balanced
An equation represents two equal quantities.
Whatever valid operation we perform on one side must also be performed on the other side.
For example:
2x + 6 = 18
Subtract 6 from both sides:
2x + 6 − 6 = 18 − 6
2x = 12
Then divide both sides by 2:
2x/2 = 12/2
x = 6
The equation remains balanced throughout the process.
Inverse Operations
The main inverse operations are:
| Operation | Inverse |
|---|---|
| Addition | Subtraction |
| Subtraction | Addition |
| Multiplication | Division |
| Division | Multiplication |
Multi-step equations usually require us to use more than one of these.
Undo Operations in Reverse Order
Consider:
5x + 8 = 33
To create the left side:
- x was multiplied by 5.
- Then 8 was added.
To solve, reverse that order:
- subtract 8
- divide by 5
5x + 8 = 33
5x = 25
x = 5
This is sometimes described as undoing the operations from the outside inward.
Why Reverse Order?
Suppose:
3x + 4 = 19
The expression 3x + 4 means:
Start with x.
↓
Multiply by 3.
↓
Add 4.
To get back to x:
Subtract 4.
↓
Divide by 3.
This reverses the original sequence.
A Reliable Solving Strategy
For many multi-step equations, use this sequence:
Simplify → Move → Isolate → Check
Simplify
- distribute if necessary
- combine like terms
Move
- move variable terms to one side if needed
- move constants to the other side
Isolate
- use inverse operations until the variable is alone
Check
- substitute the answer into the original equation
This method works for a wide range of equations.
Two-Step Addition Equation
Solve:
4x + 7 = 31
Subtract 7:
4x = 24
Divide by 4:
x = 6
Check:
4(6) + 7 = 31
24 + 7 = 31
31 = 31
Therefore:
x = 6
Two-Step Subtraction Equation
Solve:
5x − 9 = 26
Add 9:
5x = 35
Divide by 5:
x = 7
Check:
5(7) − 9 = 26
35 − 9 = 26
26 = 26
Correct.
Division and Addition
Solve:
x/4 + 3 = 8
Subtract 3:
x/4 = 5
Multiply by 4:
x = 20
Check:
20/4 + 3 = 8
5 + 3 = 8
Correct.
Division and Subtraction
Solve:
x/5 − 4 = 3
Add 4:
x/5 = 7
Multiply by 5:
x = 35
Check:
35/5 − 4 = 3
7 − 4 = 3
Correct.
Negative Coefficients
Solve:
−3x + 5 = 20
Subtract 5:
−3x = 15
Divide by −3:
x = −5
Check:
−3(−5) + 5 = 20
15 + 5 = 20
Correct.
Negative Constants
Solve:
4x − 6 = −18
Add 6:
4x = −12
Divide by 4:
x = −3
Check:
4(−3) − 6 = −18
−12 − 6 = −18
Correct.
Equations with Several Terms
Sometimes we must simplify before isolating the variable.
Consider:
3x + 5 + 2x = 25
Combine like terms:
5x + 5 = 25
Subtract 5:
5x = 20
Divide by 5:
x = 4
Check:
3(4) + 5 + 2(4) = 25
12 + 5 + 8 = 25
25 = 25
Combining Like Terms
Like terms have the same variable raised to the same power.
For example:
3x + 5x = 8x
and:
7 − 2 = 5
But:
3x + 5
cannot be combined because one term contains x and the other does not.
Worked Example: Combine First
Solve:
7x − 3x + 6 = 30
Combine like terms:
4x + 6 = 30
Subtract 6:
4x = 24
Divide by 4:
x = 6
The Distributive Property
Some equations contain parentheses.
For example:
3(x + 4) = 24
We can use the distributive property:
3x + 12 = 24
Subtract 12:
3x = 12
Divide by 3:
x = 4
Distributive Property
The distributive property states:
a(b + c) = ab + ac
and:
a(b − c) = ab − ac
For example:
4(x + 3)
becomes:
4x + 12
Similarly:
5(x − 2)
becomes:
5x − 10
Worked Example: Distribution
Solve:
2(x + 5) = 18
Distribute:
2x + 10 = 18
Subtract 10:
2x = 8
Divide by 2:
x = 4
Check using the original equation:
2(4 + 5) = 18
2(9) = 18
18 = 18
Correct.
Negative Distribution
Be especially careful when distributing a negative number.
Solve:
−2(x − 3) = 10
Distribute −2:
−2x + 6 = 10
Subtract 6:
−2x = 4
Divide by −2:
x = −2
Check:
−2(−2 − 3) = 10
−2(−5) = 10
10 = 10
Correct.
Variables on Both Sides
Some equations contain variables on both sides.
For example:
5x + 2 = 3x + 14
We first collect variable terms on one side.
Subtract 3x from both sides:
2x + 2 = 14
Subtract 2:
2x = 12
Divide by 2:
x = 6
Which Variable Term Should You Move?
Consider:
2x + 15 = 7x
We could subtract either 2x or 7x from both sides.
It is often convenient to move the smaller variable term:
Subtract 2x:
15 = 5x
Divide by 5:
3 = x
Usually we write:
x = 3
Choosing a convenient side can reduce negative numbers, but either correct method works.
Worked Example: Variables on Both Sides
Solve:
8x − 5 = 3x + 20
Subtract 3x:
5x − 5 = 20
Add 5:
5x = 25
Divide by 5:
x = 5
Check:
Left side:
8(5) − 5 = 35
Right side:
3(5) + 20 = 35
Therefore:
35 = 35
Correct.
Variables and Negative Numbers
Solve:
3x + 10 = 7x − 6
Subtract 3x:
10 = 4x − 6
Add 6:
16 = 4x
Divide by 4:
x = 4
Check:
LHS:
3(4) + 10 = 22
RHS:
7(4) − 6 = 22
Correct.
Parentheses on Both Sides
Solve:
2(x + 3) = 4(x − 2)
Distribute:
2x + 6 = 4x − 8
Subtract 2x:
6 = 2x − 8
Add 8:
14 = 2x
Divide by 2:
x = 7
Check:
LHS:
2(7 + 3) = 20
RHS:
4(7 − 2) = 20
Correct.
Simplify Both Sides First
Consider:
3(x + 2) + x = 2x + 14
Distribute:
3x + 6 + x = 2x + 14
Combine like terms:
4x + 6 = 2x + 14
Subtract 2x:
2x + 6 = 14
Subtract 6:
2x = 8
Divide by 2:
x = 4
A More Complex Example
Solve:
4(2x − 3) + 5 = 3x + 18
Distribute:
8x − 12 + 5 = 3x + 18
Combine constants:
8x − 7 = 3x + 18
Subtract 3x:
5x − 7 = 18
Add 7:
5x = 25
Divide by 5:
x = 5
Check:
LHS:
4(2(5) − 3) + 5
= 4(10 − 3) + 5
= 4(7) + 5
= 33
RHS:
3(5) + 18
= 15 + 18
= 33
Correct.
Equations Containing Fractions
Fractions can make an equation look more difficult, but the same principles apply.
Solve:
x/3 + 4 = 10
Subtract 4:
x/3 = 6
Multiply by 3:
x = 18
Clearing Fractions
Sometimes several fractions appear in an equation.
For example:
x/2 + x/3 = 10
The denominators are 2 and 3.
Their lowest common multiple is:
6
Multiply every term by 6:
6(x/2) + 6(x/3) = 6(10)
Simplify:
3x + 2x = 60
5x = 60
x = 12
Check:
12/2 + 12/3 = 10
6 + 4 = 10
Correct.
Equations with Decimals
Solve:
0.5x + 3 = 9
Subtract 3:
0.5x = 6
Divide by 0.5:
x = 12
Check:
0.5(12) + 3 = 9
6 + 3 = 9
Correct.
Clearing Decimals
Sometimes it is convenient to multiply the entire equation by a power of 10.
Consider:
0.2x + 1.5 = 3.1
Multiply every term by 10:
2x + 15 = 31
Subtract 15:
2x = 16
Divide by 2:
x = 8
Check:
0.2(8) + 1.5 = 3.1
1.6 + 1.5 = 3.1
Correct.
Equations with a Fractional Coefficient
Solve:
(3/4)x + 2 = 11
Subtract 2:
(3/4)x = 9
Multiply by the reciprocal 4/3:
x = 9 × 4/3
x = 12
Check:
(3/4)(12) + 2 = 11
9 + 2 = 11
Correct.
One Solution
Most equations in this topic have exactly one solution.
Example:
3x + 4 = 19
Solving gives:
x = 5
There is exactly one value that makes the equation true.
No Solution
Some equations have:
no solution
Consider:
3(x + 2) = 3x + 10
Distribute:
3x + 6 = 3x + 10
Subtract 3x:
6 = 10
This is false.
Therefore:
no value of x can satisfy the equation
So the equation has:
no solution
Infinitely Many Solutions
Consider:
4(x + 2) = 4x + 8
Distribute:
4x + 8 = 4x + 8
Subtract 4x:
8 = 8
This statement is always true.
Therefore every permissible value of x satisfies the equation.
The equation has:
infinitely many solutions
Recognizing the Three Outcomes
After simplifying, you may obtain:
Variable remains
2x = 10
This gives:
one solution
False statement
4 = 9
This means:
no solution
True statement
7 = 7
This means:
infinitely many solutions
Verifying Solutions
Checking becomes especially important with multi-step equations because there are more opportunities for mistakes.
Suppose:
2(3x − 1) = 22
Solve:
6x − 2 = 22
6x = 24
x = 4
Now check using the original equation:
2(3(4) − 1) = 22
2(12 − 1) = 22
2(11) = 22
22 = 22
The solution is verified.
Why Check the Original Equation?
Suppose you make a mistake while simplifying.
If you check only your simplified equation, you might not detect that mistake.
Substituting into the:
original equation
tests whether the proposed solution truly satisfies the equation you were asked to solve.
Explaining Your Reasoning
A strong solution does more than give the answer.
Consider:
6x − 8 = 28
A clear explanation would be:
"First, I add 8 to both sides because addition is the inverse of subtracting 8. This gives 6x = 36. Then I divide both sides by 6 because x is multiplied by 6. This gives x = 6. Finally, I substitute 6 into the original equation and obtain 28 = 28, so the solution is correct."
This explains:
- the operation
- the inverse operation
- why both sides are changed
- the solution
- the verification
Writing Clear Algebraic Work
Instead of jumping directly from:
5x + 7 = 32
to:
x = 5
show the important steps:
5x + 7 = 32
5x = 25
x = 5
Clear working makes it easier to:
- explain reasoning
- find mistakes
- check calculations
- communicate mathematics
Multi-Step Equations from Word Problems
Equations can model real situations.
Example
A taxi charges a fixed fee of $4 plus $3 per kilometer.
A journey costs $25.
Let x be the number of kilometers.
Equation:
3x + 4 = 25
Subtract 4:
3x = 21
Divide by 3:
x = 7
The journey was:
7 km
Worked Word Problem: Shopping
A person buys 4 identical notebooks and pays a $6 delivery charge.
The total cost is $38.
Let x represent the cost of one notebook.
4x + 6 = 38
Subtract 6:
4x = 32
Divide by 4:
x = 8
Each notebook costs:
$8
Worked Word Problem: Temperature
A temperature in Fahrenheit can be related to Celsius using:
F = (9/5)C + 32
Suppose:
F = 68
Then:
68 = (9/5)C + 32
Subtract 32:
36 = (9/5)C
Multiply by 5/9:
C = 20
Therefore:
68°F = 20°C
Multi-Step Equations in Science
Many scientific calculations involve multi-step equations.
For example:
v = u + at
Suppose:
v = 25 m/s
u = 5 m/s
t = 4 s
Then:
25 = 5 + 4a
Subtract 5:
20 = 4a
Divide by 4:
a = 5 m/s²
The same algebraic methods apply.
Another Science Example
Suppose:
E = mcΔT
and:
E = 8400 J
m = 2 kg
c = 420 J/(kg·°C)
Then:
8400 = 2(420)ΔT
8400 = 840ΔT
Divide by 840:
ΔT = 10°C
Solving scientific equations relies heavily on algebraic isolation.
Reverse Operations and Order of Operations
When evaluating an expression, we use the order of operations.
For example:
3x + 5
means:
- multiply x by 3
- add 5
When solving:
3x + 5 = 20
we undo those operations in reverse:
- subtract 5
- divide by 3
So solving can be thought of as:
reversing the construction of the expression
A Useful Flow for More Complex Equations
For equations such as:
3(2x + 5) − 4 = 2x + 27
use:
Distribute
↓
Combine like terms
↓
Collect variable terms
↓
Collect constants
↓
Isolate the variable
↓
Check
Let's apply it.
Distribute:
6x + 15 − 4 = 2x + 27
Combine:
6x + 11 = 2x + 27
Subtract 2x:
4x + 11 = 27
Subtract 11:
4x = 16
Divide by 4:
x = 4
Check:
LHS:
3(2(4) + 5) − 4
= 3(13) − 4
= 39 − 4
= 35
RHS:
2(4) + 27
= 35
Correct.
Common Mistakes
Mistake: Undoing operations in the wrong order.
For:
3x + 6 = 18
remove the +6 before dividing by 3.
Mistake: Performing an operation on only one side.
Equality must be maintained.
Mistake: Forgetting to distribute to every term.
Incorrect:
3(x + 4) = 3x + 4
Correct:
3(x + 4) = 3x + 12
Mistake: Incorrectly distributing a negative.
Correct:
−2(x − 5) = −2x + 10
Mistake: Combining unlike terms.
3x + 5
cannot become:
8x
because 3x and 5 are unlike terms.
Mistake: Moving terms without understanding the operation.
Terms do not magically "cross the equals sign and change signs."
For example:
3x + 7 = 19
becomes:
3x + 7 − 7 = 19 − 7
The −7 comes from subtracting 7 from both sides.
Mistake: Losing negative signs.
Always carry signs carefully through each step.
Mistake: Stopping too early.
If you reach:
4x = 20
the variable is not yet isolated.
Continue:
x = 5
Mistake: Forgetting to check.
Substitute the final value into the original equation.
Key Terms
Multi-step equation — equation requiring more than one operation to solve.
Variable — letter or symbol representing an unknown or changing value.
Solution — value that makes an equation true.
Inverse operations — operations that undo one another.
Isolate — get the variable alone on one side of an equation.
Like terms — terms containing the same variables raised to the same powers.
Combine like terms — simplify an expression by adding or subtracting coefficients of like terms.
Distributive property — property represented by a(b + c) = ab + ac.
Coefficient — numerical factor multiplying a variable.
Constant — number without a variable.
Substitution — replacing a variable with a particular value.
Verification — checking that a solution makes the original equation true.
Equivalent equations — equations having the same solution set.
Identity — equation true for every permissible value of the variable.
No solution — situation in which no value makes an equation true.
Reciprocal — number obtained by reversing the numerator and denominator of a nonzero fraction.
Key Takeaways
- Multi-step equations require several operations to isolate a variable.
- The fundamental goal is to get the variable alone.
- Equality must be maintained throughout the solution.
- Inverse operations undo operations affecting the variable.
- Operations are often undone in reverse order.
- Simplify expressions before trying to isolate the variable.
- Use the distributive property to remove parentheses when needed.
- Combine like terms whenever possible.
- Variables appearing on both sides can be collected onto one side.
- Constants can then be collected onto the opposite side.
- Fractions can sometimes be removed by multiplying by a common denominator.
- Decimals can sometimes be removed by multiplying by a power of 10.
- Negative signs require careful attention.
- A multi-step equation may have one solution, no solution, or infinitely many solutions.
- A false statement such as 4 = 9 indicates no solution.
- A true statement such as 7 = 7 after the variable disappears indicates infinitely many solutions.
- Solutions should be checked by substitution into the original equation.
- Clear algebraic steps make mathematical reasoning easier to communicate and verify.
- Multi-step equations are important for solving mathematical models and rearranging equations in science.
- Systematic methods reduce mistakes and make even complicated equations manageable.
Check Your Understanding
- What is a multi-step equation?
- What does it mean to isolate a variable?
- Why are operations often undone in reverse order when solving equations?
- Solve: 3x + 5 = 20
- Solve: 4x − 7 = 29
- Solve: x/5 + 3 = 9
- Solve: −3x + 8 = 23
- Solve: 5x + 7 + 2x = 42
- Solve: 3(x + 4) = 27
- Solve: −2(x − 5) = 18
- Solve: 5x + 3 = 2x + 21
- Solve: 7x − 4 = 3x + 20
- Solve: 2(x + 5) = 4(x − 1)
- Solve: x/2 + x/3 = 15
- Solve: 0.4x + 2.6 = 8.2
- Determine whether 4(x + 2) = 4x + 8 has one solution, no solution, or infinitely many solutions.
- Determine whether 3(x + 1) = 3x + 8 has one solution, no solution, or infinitely many solutions.
- A taxi charges $5 plus $4 per kilometer. The total fare is $37. Write and solve an equation to determine the distance traveled.
- A student solves 4x + 8 = 28 and obtains x = 9. Use substitution to determine whether the answer is correct and explain the error.
- Explain a systematic method you could use to solve an equation containing parentheses, like terms, and variables on both sides.
4. Equations with Variables on Both Sides
Learning outcomes
- I can simplify equations with variables on both sides.
- I can collect variable terms onto one side of an equation.
- I can use balancing techniques to solve equations efficiently.
- I can identify equations with one solution, no solution, or infinitely many solutions.
- I can verify solutions using substitution.
Equations with Variables on Both Sides
Some equations contain the same variable on both sides of the equals sign.
For example:
5x + 3 = 2x + 18
To solve this equation, we need to collect the variable terms onto one side and the constants onto the other.
Subtract 2x from both sides:
3x + 3 = 18
Subtract 3:
3x = 15
Divide by 3:
x = 5
The key idea is the same as with simpler equations:
keep the equation balanced while isolating the variable.
Why Variables Can Appear on Both Sides
An equation compares two expressions.
There is no requirement that the variable appear on only one side.
For example:
7x − 4 = 3x + 20
means that:
7x − 4
and:
3x + 20
have the same value for the solution of x.
Our goal is to determine which value makes that equality true.
The Main Strategy
For many equations with variables on both sides, use:
Simplify → Collect variables → Collect constants → Isolate → Check
Simplify
- distribute if necessary
- combine like terms
Collect variables
Move the variable terms to one side.
Collect constants
Move constants to the opposite side.
Isolate
Divide or multiply as necessary to get the variable alone.
Check
Substitute the solution into the original equation.
Collecting Variable Terms
Consider:
6x + 5 = 2x + 21
We can remove 2x from the right by subtracting 2x from both sides:
6x − 2x + 5 = 2x − 2x + 21
Simplify:
4x + 5 = 21
The variable now appears on only one side.
Finish the Equation
Continue:
4x + 5 = 21
Subtract 5:
4x = 16
Divide by 4:
x = 4
Check:
Left side:
6(4) + 5 = 29
Right side:
2(4) + 21 = 29
Therefore:
29 = 29
So:
x = 4
Terms Do Not Really "Move"
You may sometimes hear:
"Move 2x to the other side and change its sign."
This shortcut can be useful, but it hides what is actually happening.
From:
6x + 5 = 2x + 21
we do not magically move 2x.
We:
subtract 2x from both sides
That preserves equality.
Understanding this balancing process helps prevent sign errors.
Choosing Which Variable Term to Remove
Consider:
8x + 3 = 5x + 24
It is convenient to subtract 5x:
3x + 3 = 24
This keeps the coefficient positive.
Then:
3x = 21
x = 7
We could instead subtract 8x, but this would create a negative coefficient.
Both approaches are mathematically valid.
An Alternative Correct Method
Start with:
8x + 3 = 5x + 24
Subtract 8x:
3 = −3x + 24
Subtract 24:
−21 = −3x
Divide by −3:
x = 7
Same solution.
Therefore:
you can collect variables on either side
as long as you maintain equality correctly.
Efficient Solving
It is often easier to move the smaller variable term toward the larger variable term.
For:
9x + 2 = 4x + 27
subtract 4x:
5x + 2 = 27
This avoids negative coefficients.
Subtract 2:
5x = 25
Divide by 5:
x = 5
Efficiency matters, but correctness matters more.
Worked Example
Solve:
7x − 8 = 3x + 20
Subtract 3x:
4x − 8 = 20
Add 8:
4x = 28
Divide by 4:
x = 7
Check:
LHS:
7(7) − 8 = 41
RHS:
3(7) + 20 = 41
Therefore:
41 = 41
Correct.
Worked Example with Negative Numbers
Solve:
2x + 11 = 5x − 4
Subtract 2x:
11 = 3x − 4
Add 4:
15 = 3x
Divide by 3:
x = 5
Check:
LHS:
2(5) + 11 = 21
RHS:
5(5) − 4 = 21
Correct.
Negative Coefficients
Solve:
−3x + 7 = 2x − 18
Subtract 2x:
−5x + 7 = −18
Subtract 7:
−5x = −25
Divide by −5:
x = 5
Check:
LHS:
−3(5) + 7 = −8
RHS:
2(5) − 18 = −8
Correct.
Variables with Several Terms
Sometimes we must combine like terms before collecting the variables.
Solve:
3x + 4 + 2x = 7x − 10
Combine the left-side variable terms:
5x + 4 = 7x − 10
Subtract 5x:
4 = 2x − 10
Add 10:
14 = 2x
Divide by 2:
x = 7
Simplify First
A common mistake is trying to move terms before simplifying.
Consider:
4x + 3x − 8 = 2x + 17
First combine:
7x − 8 = 2x + 17
Then subtract 2x:
5x − 8 = 17
Add 8:
5x = 25
Divide by 5:
x = 5
Equations with Parentheses
Parentheses should usually be removed before collecting variable terms.
Consider:
3(x + 2) = 2x + 10
Distribute:
3x + 6 = 2x + 10
Subtract 2x:
x + 6 = 10
Subtract 6:
x = 4
Parentheses on Both Sides
Solve:
2(x + 5) = 3(x + 1)
Distribute:
2x + 10 = 3x + 3
Subtract 2x:
10 = x + 3
Subtract 3:
7 = x
Therefore:
x = 7
Check:
LHS:
2(7 + 5) = 24
RHS:
3(7 + 1) = 24
Correct.
Negative Distribution
Solve:
−2(x − 4) = 3x − 7
Distribute:
−2x + 8 = 3x − 7
Subtract 3x:
−5x + 8 = −7
Subtract 8:
−5x = −15
Divide by −5:
x = 3
Check:
LHS:
−2(3 − 4) = 2
RHS:
3(3) − 7 = 2
Correct.
A More Complex Equation
Solve:
3(2x + 1) − 4 = 2(x + 6)
Distribute:
6x + 3 − 4 = 2x + 12
Combine:
6x − 1 = 2x + 12
Subtract 2x:
4x − 1 = 12
Add 1:
4x = 13
Divide by 4:
x = 13/4
or:
x = 3.25
Not every equation produces an integer solution.
Three Possible Outcomes
Equations with variables on both sides can produce three important outcomes:
one solution
no solution
infinitely many solutions
Understanding why these occur is an important part of algebra.
One Solution
Consider:
5x + 4 = 2x + 19
Subtract 2x:
3x + 4 = 19
Subtract 4:
3x = 15
Divide by 3:
x = 5
The variable remains during the solving process and eventually produces one specific value.
Therefore the equation has:
one solution
No Solution
Consider:
4x + 3 = 4x + 9
Subtract 4x from both sides:
3 = 9
This statement is false.
No value of x can make:
3 = 9
true.
Therefore:
the equation has no solution
Why No Solution Occurs
Consider:
4x + 3 = 4x + 9
Both sides contain exactly the same variable term:
4x
But one side is always 6 greater than the other.
For any x:
Left side:
4x + 3
Right side:
4x + 9
They can never be equal.
Infinitely Many Solutions
Consider:
3x + 6 = 3(x + 2)
Distribute the right side:
3x + 6 = 3x + 6
Subtract 3x:
6 = 6
This statement is always true.
Therefore:
every permissible value of x is a solution
The equation has:
infinitely many solutions
Why Infinitely Many Solutions Occur
Consider:
2(x + 4) = 2x + 8
The left side simplifies to:
2x + 8
So the equation becomes:
2x + 8 = 2x + 8
Both sides are actually the same expression.
Therefore they are equal for every value of x.
Recognizing the Outcome
After simplifying and collecting terms:
If you get:
x = a number
there is:
one solution
If the variable disappears and you get a false statement:
5 = 12
there is:
no solution
If the variable disappears and you get a true statement:
8 = 8
there are:
infinitely many solutions
Worked Example: Determine the Number of Solutions
Solve:
5x − 2 = 5x − 2
Subtract 5x:
−2 = −2
This is always true.
Therefore:
infinitely many solutions
Worked Example: No Solution
Solve:
7x + 4 = 7x − 3
Subtract 7x:
4 = −3
False.
Therefore:
no solution
Worked Example: One Solution
Solve:
7x + 4 = 5x + 18
Subtract 5x:
2x + 4 = 18
Subtract 4:
2x = 14
Divide by 2:
x = 7
Therefore:
one solution: x = 7
The Graph Connection
A linear equation with variables on both sides can also be interpreted as asking:
When do two linear expressions have the same value?
For example:
2x + 3 = x + 7
can be represented by:
y = 2x + 3
and:
y = x + 7
The solution occurs where the two lines intersect.
If they intersect once:
one solution
If they are parallel:
no solution
If they are the same line:
infinitely many solutions
One Intersection
Consider:
2x + 3 = x + 7
Solve:
x + 3 = 7
x = 4
The corresponding lines intersect once.
Therefore the equation has:
one solution
Parallel Lines and No Solution
Consider:
2x + 3 = 2x + 8
Both expressions have the same coefficient of x but different constants.
Graphically:
y = 2x + 3
and:
y = 2x + 8
have the same slope but different y-intercepts.
They are parallel.
They never intersect.
Therefore:
no solution
Same Line and Infinite Solutions
Consider:
2(x + 3) = 2x + 6
Both sides simplify to:
2x + 6
Graphically, both expressions represent the same line.
Every point on one is also on the other.
Therefore:
infinitely many solutions
Fractions with Variables on Both Sides
Solve:
x/2 + 3 = x/4 + 8
One approach is to clear the fractions.
The lowest common denominator is 4.
Multiply every term by 4:
2x + 12 = x + 32
Subtract x:
x + 12 = 32
Subtract 12:
x = 20
Check:
LHS:
20/2 + 3 = 13
RHS:
20/4 + 8 = 13
Correct.
Decimals with Variables on Both Sides
Solve:
0.5x + 4 = 0.2x + 10
Subtract 0.2x:
0.3x + 4 = 10
Subtract 4:
0.3x = 6
Divide by 0.3:
x = 20
Check:
LHS:
0.5(20) + 4 = 14
RHS:
0.2(20) + 10 = 14
Correct.
Clearing Decimals First
We could also solve:
0.5x + 4 = 0.2x + 10
by multiplying every term by 10:
5x + 40 = 2x + 100
Subtract 2x:
3x + 40 = 100
Subtract 40:
3x = 60
x = 20
Both methods are valid.
Variables in Numerators
Solve:
(x + 2)/3 = (x − 2)/2
Multiply every term by 6:
2(x + 2) = 3(x − 2)
Distribute:
2x + 4 = 3x − 6
Subtract 2x:
4 = x − 6
Add 6:
x = 10
Check:
LHS:
(10 + 2)/3 = 4
RHS:
(10 − 2)/2 = 4
Correct.
Verification by Substitution
After finding a solution, substitute it into the original equation.
Suppose:
4x + 5 = 2x + 17
Solving gives:
x = 6
Check the left-hand side:
4(6) + 5
= 24 + 5
= 29
Check the right-hand side:
2(6) + 17
= 12 + 17
= 29
Therefore:
LHS = RHS
and x = 6 is verified.
Why Verification Matters
Equations with variables on both sides involve several opportunities for mistakes:
- incorrect subtraction
- lost negative signs
- incorrect distribution
- combining unlike terms
- arithmetic errors
Substitution tests whether the final answer actually satisfies the original equation.
Worked Example: Catching an Error
A student claims that:
5x + 4 = 3x + 18
has solution:
x = 5
Check:
LHS:
5(5) + 4 = 29
RHS:
3(5) + 18 = 33
Since:
29 ≠ 33
x = 5 is incorrect.
Solve correctly:
5x + 4 = 3x + 18
Subtract 3x:
2x + 4 = 18
Subtract 4:
2x = 14
Divide by 2:
x = 7
Check:
39 = 39
Correct.
Word Problem: Two Pricing Plans
Company A charges:
$15 + $4 per unit
Company B charges:
$27 + $2 per unit
When will the costs be equal?
Let x be the number of units.
4x + 15 = 2x + 27
Subtract 2x:
2x + 15 = 27
Subtract 15:
2x = 12
Divide by 2:
x = 6
The plans cost the same at:
6 units
Check:
Company A:
4(6) + 15 = $39
Company B:
2(6) + 27 = $39
Word Problem: Two Moving Objects
One object is:
10 m
ahead and moves at:
3 m/s
Another starts at 0 m and moves at:
5 m/s
Their positions can be represented as:
First object:
3t + 10
Second object:
5t
They meet when:
3t + 10 = 5t
Subtract 3t:
10 = 2t
Therefore:
t = 5 s
After 5 seconds, both are at:
25 m
Equations in Geometry
Suppose two sides of a shape must have equal lengths.
One side is:
3x + 5
The other is:
5x − 9
Set them equal:
3x + 5 = 5x − 9
Subtract 3x:
5 = 2x − 9
Add 9:
14 = 2x
Therefore:
x = 7
Equations with variables on both sides frequently appear in geometry.
Equations in Science
Suppose two objects have position equations:
d₁ = 4t + 6
d₂ = 7t − 9
To find when they have the same position:
4t + 6 = 7t − 9
Subtract 4t:
6 = 3t − 9
Add 9:
15 = 3t
Therefore:
t = 5
This shows how variables-on-both-sides equations can model physical situations.
A Systematic Method
For complicated equations, follow this order:
Remove parentheses
↓
Combine like terms on each side
↓
Collect variable terms on one side
↓
Collect constants on the other side
↓
Divide by the variable's coefficient
↓
Check the solution
↓
Classify the equation
This process keeps the algebra organized.
Full Worked Example
Solve:
4(2x − 3) + 5 = 3(x + 4) + 8
Distribute:
8x − 12 + 5 = 3x + 12 + 8
Combine:
8x − 7 = 3x + 20
Subtract 3x:
5x − 7 = 20
Add 7:
5x = 27
Divide by 5:
x = 27/5
or:
x = 5.4
Check using the original equation.
LHS:
4(2(5.4) − 3) + 5
= 4(10.8 − 3) + 5
= 4(7.8) + 5
= 36.2
RHS:
3(5.4 + 4) + 8
= 3(9.4) + 8
= 36.2
Correct.
Common Mistakes
Mistake: Changing the sign of a term without performing an operation on both sides.
Instead of saying:
"Move 3x and change its sign,"
understand that you are:
subtracting 3x from both sides.
Mistake: Combining terms across the equals sign.
In:
5x + 2 = 3x + 10
you cannot combine 5x and 3x immediately because they are on different sides.
First perform the same operation on both sides.
Mistake: Combining unlike terms.
4x + 7
cannot become:
11x
Mistake: Forgetting to distribute.
3(x + 5)
equals:
3x + 15
not:
3x + 5
Mistake: Losing a negative sign during distribution.
−2(x − 4)
equals:
−2x + 8
Mistake: Assuming every equation has exactly one solution.
Variables-on-both-sides equations may have:
- one solution
- no solution
- infinitely many solutions
Mistake: Thinking 5 = 5 means x = 5.
If the variable disappears and the result is:
5 = 5
the equation is true for every permissible value.
Therefore there are:
infinitely many solutions
Mistake: Thinking 5 = 8 means x has been calculated incorrectly.
If the algebra is correct and the variable disappears to produce a false statement, the equation has:
no solution
Key Terms
Variable on both sides — equation in which the same variable occurs on both sides of the equals sign.
Variable term — term containing a variable.
Constant — number without a variable.
Coefficient — number multiplying a variable.
Like terms — terms containing the same variables raised to the same powers.
Collect terms — combine or move terms so similar terms appear together.
Inverse operations — operations that undo one another.
Balance — maintaining equality by performing equivalent operations on both sides.
Isolate — get the variable alone on one side.
Distributive property — property represented by a(b + c) = ab + ac.
Substitution — replacing a variable with a particular value.
Verification — checking that a solution satisfies the original equation.
One solution — exactly one value satisfies the equation.
No solution — no value satisfies the equation.
Infinitely many solutions — every permissible value satisfies the equation.
Identity — equation that is true for every permissible value of its variable.
Equivalent equations — equations with the same solution set.
Key Takeaways
- Variables can appear on both sides of an equation.
- The goal is still to isolate the variable.
- Variable terms should usually be collected onto one side.
- Constants should usually be collected onto the other side.
- Performing the same operation on both sides maintains equality.
- Terms do not literally jump across the equals sign and change signs.
- You can collect the variable on either side of the equation.
- Choosing the side that produces a positive coefficient can make calculations easier.
- Simplify each side before collecting terms.
- Use the distributive property to remove parentheses.
- Combine like terms carefully.
- Negative coefficients and constants follow the same algebraic rules.
- Fractions and decimals do not change the fundamental solving process.
- Clearing fractions or decimals can sometimes simplify an equation.
- Equations with variables on both sides can have one solution, no solution, or infinitely many solutions.
- A false statement after the variable disappears indicates no solution.
- A true statement after the variable disappears indicates infinitely many solutions.
- One solution corresponds graphically to two lines intersecting once.
- No solution corresponds to distinct parallel lines.
- Infinitely many solutions corresponds to the same line represented in two equivalent forms.
- Substitution into the original equation is an effective way to verify a solution.
- These equations can model comparisons such as pricing plans, motion, geometry, and scientific relationships.
Check Your Understanding
- What does it mean to collect variable terms onto one side of an equation?
- Why must the same operation be performed on both sides?
- Solve: 5x + 3 = 2x + 18
- Solve: 8x − 7 = 3x + 23
- Solve: 2x + 14 = 6x − 10
- Solve: −3x + 8 = 2x − 17
- Solve: 3(x + 4) = 2x + 17
- Solve: 2(x + 5) = 4(x − 2)
- Solve: 5x + 3x − 4 = 2x + 20
- Solve: x/2 + 4 = x/4 + 9
- Solve: 0.6x + 5 = 0.2x + 13
- Determine whether 4x + 7 = 4x + 12 has one solution, no solution, or infinitely many solutions.
- Determine whether 5x − 3 = 5x − 3 has one solution, no solution, or infinitely many solutions.
- Determine whether 3(x + 2) = 3x + 6 has one solution, no solution, or infinitely many solutions.
- Explain why 6 = 11 appearing after simplification means that an equation has no solution.
- Explain why 8 = 8 appearing after simplification can mean that an equation has infinitely many solutions.
- Verify by substitution that x = 4 solves 5x + 2 = 3x + 10.
- Two companies charge 5x + 10 and 3x + 24 dollars. Find the value of x at which their costs are equal.
- Explain the connection between one solution, no solution, and infinitely many solutions and the intersections of two straight lines.
- Describe a systematic method for solving an equation containing parentheses, like terms, and variables on both sides.
5. Real-World Equation Problems
Learning outcomes
- I can translate word problems into equations.
- I can identify unknown quantities in real-world situations.
- I can solve equations to answer practical questions.
- I can interpret solutions within context.
- I can communicate mathematical reasoning clearly.
Real-World Equation Problems
Equations allow us to turn real situations into mathematics. They can be used to represent questions involving money, distance, time, temperature, measurements, science, business, and many other situations.
A typical process is:
real-world situation → define a variable → write an equation → solve → interpret the answer
For example:
A taxi charges a $5 starting fee plus $3 per kilometer. The total fare is $26. How far did the taxi travel?
Let:
x = distance traveled in kilometers
The situation becomes:
3x + 5 = 26
Solve:
3x = 21
x = 7
The mathematical solution is x = 7, but the contextual answer is:
The taxi traveled 7 km.
Mathematical Modelling
Using mathematics to represent a real situation is called:
mathematical modelling
A model simplifies a situation so that we can analyze it mathematically.
For example:
C = 4x + 10
might represent a cost consisting of:
- $10 fixed fee
- $4 for each item
The equation allows us to predict costs or determine unknown quantities.
The Main Problem-Solving Process
A reliable strategy is:
Understand → Define → Translate → Solve → Check → Interpret
Understand
Determine what the problem is describing.
Define
Choose a variable for the unknown quantity.
Translate
Turn the information into an equation.
Solve
Use algebra to isolate the variable.
Check
Determine whether the solution satisfies the equation.
Interpret
Explain what the answer means in the original situation.
Identifying the Unknown
Before writing an equation, determine:
What am I trying to find?
Suppose:
"A student has $18 and wants to buy a game costing $47. How much more money does the student need?"
The unknown is:
additional money needed
Let:
x = additional money needed
Then:
18 + x = 47
Defining Variables Clearly
A variable should have a clear meaning.
Instead of writing only:
x = unknown
write:
x = number of kilometers traveled
or:
x = cost of one ticket in dollars
or:
x = number of students
This makes the mathematical reasoning easier to understand.
Translating Words into Mathematics
Certain words often suggest mathematical operations.
| Words or phrases | Possible operation |
|---|---|
| sum, total, increased by, more than | + |
| difference, decreased by, less than | − |
| product, times, twice, triple | × |
| quotient, divided by, shared equally | ÷ |
| equals, is, gives, results in | = |
These are useful clues, but always understand the complete sentence rather than translating individual words mechanically.
Addition Problems
Example:
"A number increased by 8 is 23."
Let:
x = the number
Translate:
x + 8 = 23
Solve:
x = 15
Check:
15 + 8 = 23
Therefore:
The number is 15.
Subtraction Problems
Example:
"A number decreased by 12 is 30."
Let:
x = the number
Equation:
x − 12 = 30
Add 12:
x = 42
Therefore:
The number is 42.
Multiplication Problems
Example:
"Six identical notebooks cost $42. What is the cost of one notebook?"
Let:
x = cost of one notebook
Equation:
6x = 42
Divide by 6:
x = 7
Therefore:
Each notebook costs $7.
Division Problems
Example:
"A group shares 72 candies equally among 8 students. How many candies does each student receive?"
Let:
x = candies received by each student
Equation:
8x = 72
Solve:
x = 9
Therefore:
Each student receives 9 candies.
Translating Two-Step Situations
Many real situations involve more than one operation.
Example:
"A gym charges a $20 registration fee plus $15 per month. A customer has paid $95. How many months has the customer been a member?"
Let:
m = number of months
Monthly cost:
15m
Registration fee:
20
Total:
15m + 20
Equation:
15m + 20 = 95
Subtract 20:
15m = 75
Divide by 15:
m = 5
Therefore:
The customer has been a member for 5 months.
Fixed and Variable Costs
Many real-world equations involve:
fixed cost + variable cost = total cost
For example:
A delivery company charges:
- $8 fixed fee
- $2 per kilometer
The total cost C for x kilometers is:
C = 2x + 8
If the total cost is $30:
2x + 8 = 30
2x = 22
x = 11
Therefore:
The delivery distance was 11 km.
Shopping Problems
Suppose four identical shirts plus a $12 delivery fee cost $100.
Let:
x = cost of one shirt
Equation:
4x + 12 = 100
Subtract 12:
4x = 88
Divide by 4:
x = 22
Therefore:
Each shirt costs $22.
Discount Problems
A jacket originally costs x dollars.
A $25 discount reduces the price to $80.
Equation:
x − 25 = 80
Add 25:
x = 105
Therefore:
The original price was $105.
Percentage Problems
A store discounts an item by 20%.
After the discount, the item costs $64.
If x is the original price, then 80% of the original price remains:
0.80x = 64
Divide by 0.80:
x = 80
Therefore:
The original price was $80.
Age Problems
Suppose:
"Maria is 6 years older than Sam. Maria is 20 years old. How old is Sam?"
Let:
x = Sam's age
Maria's age:
x + 6
Equation:
x + 6 = 20
Solve:
x = 14
Therefore:
Sam is 14 years old.
Age Problems with Two Relationships
Suppose:
"A father is three times his son's age. The father is 42 years old."
Let:
x = son's age
Father's age:
3x
Equation:
3x = 42
Therefore:
x = 14
The son is:
14 years old.
Distance, Speed, and Time
One important real-world relationship is:
d = vt
where:
- d = distance
- v = speed
- t = time
Suppose a car travels:
180 km
at:
60 km/h
Let t represent the travel time.
180 = 60t
Divide by 60:
t = 3
Therefore:
The journey takes 3 hours.
Travel with a Starting Distance
Two cyclists are traveling along the same route.
Cyclist A starts 12 km ahead and travels at 4 km/h.
Cyclist B starts behind but travels at 6 km/h.
Their positions after t hours are:
Cyclist A:
4t + 12
Cyclist B:
6t
They meet when their positions are equal:
4t + 12 = 6t
Subtract 4t:
12 = 2t
Therefore:
t = 6 hours
Interpreting the Travel Solution
Finding:
t = 6
is not the complete answer.
The variable represented time.
Therefore:
The cyclists meet after 6 hours.
A good answer includes:
number + unit + meaning
Temperature Problems
Temperature conversions provide useful equation problems.
The Fahrenheit-Celsius relationship is:
F = (9/5)C + 32
Suppose:
F = 86°F
Then:
86 = (9/5)C + 32
Subtract 32:
54 = (9/5)C
Multiply by 5/9:
C = 30
Therefore:
86°F = 30°C
Geometry Problems
Equations can be used to determine unknown lengths and angles.
Suppose a rectangle has:
length = x + 4
width = x
and perimeter:
32 cm
The perimeter formula is:
P = 2l + 2w
Substitute:
32 = 2(x + 4) + 2x
Distribute:
32 = 2x + 8 + 2x
Combine:
32 = 4x + 8
Subtract 8:
24 = 4x
x = 6
Therefore:
Width:
6 cm
Length:
10 cm
Check:
2(10) + 2(6)
= 20 + 12
= 32 cm
Angle Problems
Two angles form a straight line.
One angle is:
x
The other is:
2x
Angles on a straight line total:
180°
Therefore:
x + 2x = 180
3x = 180
x = 60
The angles are:
60° and 120°
Science Problems
Equations are essential for solving scientific problems.
For example:
F = ma
Suppose an object has:
mass = 8 kg
and experiences:
force = 40 N
Let a represent acceleration:
40 = 8a
Divide by 8:
a = 5 m/s²
Therefore:
The object's acceleration is 5 m/s².
Density Problems
Density is:
ρ = m/V
Suppose an object has:
ρ = 4 g/cm³
V = 15 cm³
Let m represent mass:
4 = m/15
Multiply by 15:
m = 60
Therefore:
The object's mass is 60 g.
Energy Problems
Suppose electrical energy is calculated using:
E = Pt
A device uses:
1200 J
in:
30 s
Let P represent power:
1200 = 30P
Divide by 30:
P = 40 W
Therefore:
The device's power is 40 W.
Mixture and Total Problems
Suppose a school has 480 students.
There are 40 more girls than boys.
Let:
x = number of boys
Then:
x + 40 = number of girls
Total students:
x + (x + 40) = 480
Simplify:
2x + 40 = 480
Subtract 40:
2x = 440
Divide by 2:
x = 220
Therefore:
220 boys
and:
260 girls
Check:
220 + 260 = 480
Consecutive Number Problems
Consecutive integers differ by 1.
For example:
7, 8, 9
Suppose two consecutive integers have a sum of 41.
Let:
x = smaller integer
Then the next integer is:
x + 1
Equation:
x + (x + 1) = 41
Combine:
2x + 1 = 41
Subtract 1:
2x = 40
Divide by 2:
x = 20
Therefore the integers are:
20 and 21
Consecutive Even Numbers
Consecutive even integers differ by 2.
Let the first be:
x
Then the next is:
x + 2
Suppose their sum is 54:
x + (x + 2) = 54
2x + 2 = 54
2x = 52
x = 26
Therefore:
26 and 28
Comparing Two Plans
Equations with variables on both sides are useful for comparing costs.
Company A charges:
$10 + $5 per month
Company B charges:
$22 + $3 per month
When do they cost the same?
Let:
m = number of months
Company A:
5m + 10
Company B:
3m + 22
Set them equal:
5m + 10 = 3m + 22
Subtract 3m:
2m + 10 = 22
Subtract 10:
2m = 12
m = 6
Therefore:
The plans cost the same after 6 months.
Check the Context
Substitute m = 6.
Company A:
5(6) + 10 = $40
Company B:
3(6) + 22 = $40
The answer makes sense.
Not Every Mathematical Answer Makes Sense
Solving an equation mathematically is only part of a real-world problem.
The answer must also make sense in context.
Suppose an equation gives:
x = −4
but x represents:
the number of students attending a concert
A negative number of students is impossible.
Therefore the mathematical result indicates that something about the model, assumptions, or given conditions needs to be reconsidered.
Whole-Number Restrictions
Suppose calculations show that a school needs:
3.2 buses
A school cannot normally hire 0.2 of a bus.
If every student must be transported, the practical answer may be:
4 buses
This is an example where interpreting the solution matters as much as calculating it.
Rounding in Context
Suppose:
x = 12.746 km
If the question asks for the distance to the nearest tenth:
x ≈ 12.7 km
If x represents money:
$12.746
would normally be rounded to:
$12.75
Rounding should match the context.
Units Matter
A numerical answer without units can be incomplete.
Compare:
x = 12
with:
The journey took 12 minutes.
The second answer communicates what the solution actually means.
Common units include:
- dollars
- meters
- kilometers
- seconds
- hours
- kilograms
- liters
- degrees
- square centimeters
Checking Whether an Answer Is Reasonable
After solving, ask:
Does the sign make sense?
Is the value realistic?
Are the units correct?
Does it satisfy the original equation?
Does it answer the question actually asked?
This is sometimes called a:
reasonableness check
Translating "More Than"
Be careful with phrases involving order.
"5 more than x" means:
x + 5
For example:
"Sam has 5 more books than Alex."
If Alex has x books:
Sam has x + 5 books
Translating "Less Than"
"5 less than x" means:
x − 5
For example:
"Maria is 4 years younger than David."
If David's age is x:
Maria's age = x − 4
The order of subtraction matters.
Twice and Three Times
"Twice a number" means:
2x
"Three times a number" means:
3x
"Five more than twice a number" means:
2x + 5
"Five times the sum of a number and 2" means:
5(x + 2)
The wording determines where parentheses are needed.
Parentheses in Word Problems
Compare:
three times a number plus 4
This usually means:
3x + 4
But:
three times the sum of a number and 4
means:
3(x + 4)
These expressions are not equivalent.
For x = 2:
3x + 4 = 10
while:
3(x + 4) = 18
Careful reading matters.
From Words to Equation
Consider:
"Seven less than three times a number is 20."
Let:
x = the number
Three times the number:
3x
Seven less:
3x − 7
"is 20" means:
= 20
Therefore:
3x − 7 = 20
Solve:
3x = 27
x = 9
From Equation Back to Context
Suppose we solve:
8x + 15 = 79
and obtain:
x = 8
That alone tells us nothing about the real-world meaning unless x was defined.
If:
x = number of tickets
then the answer is:
8 tickets
If:
x = number of hours
then the answer is:
8 hours
Defining the variable gives the solution meaning.
Communicating Mathematical Reasoning
A strong written solution should show:
1. Variable definition
2. Equation
3. Algebraic steps
4. Verification or reasonableness check
5. Final answer in context
For example:
"Let x be the number of kilometers traveled. The taxi charges $5 plus $3 per kilometer, so 3x + 5 = 26. Subtracting 5 gives 3x = 21. Dividing by 3 gives x = 7. Checking gives 3(7) + 5 = 26. Therefore, the taxi traveled 7 km."
This communicates both the mathematics and the reasoning.
Worked Example: Phone Plan
A phone plan charges:
$18 fixed monthly fee
plus:
$4 per GB of additional data
The bill is:
$50
How many additional gigabytes were used?
Let:
x = additional GB used
Equation:
4x + 18 = 50
Subtract 18:
4x = 32
Divide by 4:
x = 8
Check:
4(8) + 18 = 50
Therefore:
8 GB of additional data were used.
Worked Example: Saving Money
A student already has $35 and saves $12 each week.
The student wants $131.
Let:
w = number of weeks
Equation:
12w + 35 = 131
Subtract 35:
12w = 96
Divide by 12:
w = 8
Therefore:
It will take 8 weeks.
Worked Example: Perimeter
A rectangle has a length 3 cm greater than its width.
Its perimeter is 30 cm.
Let:
x = width
Then:
length = x + 3
Perimeter:
2x + 2(x + 3) = 30
Distribute:
2x + 2x + 6 = 30
4x + 6 = 30
4x = 24
x = 6
Width:
6 cm
Length:
9 cm
Check:
2(6) + 2(9) = 30
Correct.
Worked Example: Comparing Speeds
Runner A has already traveled 400 m and continues at 4 m/s.
Runner B starts at 0 m and travels at 6 m/s.
When does Runner B catch Runner A?
Let:
t = time in seconds
Runner A:
400 + 4t
Runner B:
6t
Set equal:
400 + 4t = 6t
Subtract 4t:
400 = 2t
t = 200
Therefore:
Runner B catches Runner A after 200 seconds.
Worked Example: Science Context
A substance has a density of:
2.5 g/cm³
and mass:
50 g
Use:
ρ = m/V
Substitute:
2.5 = 50/V
Multiply by V:
2.5V = 50
Divide by 2.5:
V = 20
Therefore:
The volume is 20 cm³.
Creating an Equation from a Diagram
Real-world problems may include:
- diagrams
- tables
- graphs
- maps
- measurements
Instead of relying only on words, identify relationships between the quantities shown.
For example, if two marked lengths are equal:
3x + 4
and:
5x − 10
then:
3x + 4 = 5x − 10
The diagram provides the relationship needed to construct the equation.
When a Problem Has Extra Information
Not every number in a word problem must necessarily appear in the equation.
Good problem solvers distinguish between:
relevant information
and:
irrelevant information
Ask:
Does this information help describe the relationship involving the unknown?
If not, it may not be needed.
Assumptions in Mathematical Models
Real-world equations often depend on assumptions.
For example:
d = vt
assumes that speed remains constant during the time interval.
A taxi-cost equation such as:
C = 3x + 5
assumes the price per kilometer remains constant.
A model can be mathematically correct while still being an imperfect description of reality.
Common Mistakes
Mistake: Starting calculations before defining the unknown.
First determine what the variable represents.
Mistake: Translating words mechanically.
Understand the complete relationship before choosing operations.
Mistake: Reversing subtraction.
"5 less than x" means x − 5, not 5 − x.
Mistake: Forgetting fixed costs.
A $10 fee plus $3 per item is 3x + 10, not just 3x.
Mistake: Ignoring parentheses.
"Three times the sum of x and 4" means 3(x + 4).
Mistake: Giving only x = 8.
Interpret the result using the correct units and context.
Mistake: Accepting impossible answers.
Negative people, fractional cars, or unrealistic measurements may indicate that the answer must be interpreted differently or the model reconsidered.
Mistake: Forgetting units.
Units are part of a meaningful real-world answer.
Mistake: Not checking the solution.
Substitution and a reasonableness check can reveal errors.
Key Terms
Word problem — mathematical problem presented using a real or contextual situation.
Mathematical model — mathematical representation of a real system or situation.
Variable — symbol representing an unknown or changing quantity.
Unknown quantity — value that the problem asks us to determine.
Equation — mathematical statement showing that two expressions have equal values.
Translate — convert information from words or another representation into mathematical form.
Fixed cost — cost that does not change with the number of items or amount of use.
Variable cost — cost that changes according to quantity or usage.
Constraint — condition limiting possible solutions.
Substitution — replacing a variable with a particular value.
Verification — checking whether a proposed solution satisfies the equation.
Interpretation — explaining what a mathematical result means in its original context.
Reasonableness check — deciding whether an answer is realistic and appropriate.
Unit — standard measurement attached to a quantity, such as meters, seconds, or dollars.
Assumption — condition accepted when constructing a mathematical model.
Key Takeaways
- Equations can represent real-world relationships.
- Begin by identifying the unknown quantity.
- Define variables clearly and include what they represent.
- Translate relationships into mathematical expressions before solving.
- Addition, subtraction, multiplication, and division can all appear in word problems.
- Fixed and variable quantities often produce equations such as C = mx + b.
- Multi-step equations frequently appear in cost, geometry, science, and travel problems.
- Variables on both sides can represent comparisons between two plans or moving objects.
- Parentheses are important when an operation applies to an entire expression.
- Scientific formulas are mathematical models of physical relationships.
- A mathematical solution must be interpreted within the original context.
- Include appropriate units in final answers.
- Some mathematical solutions may not be physically or practically meaningful.
- Whole-number contexts may require careful rounding.
- Checking by substitution helps verify algebraic accuracy.
- A reasonableness check determines whether the result makes sense.
- Real-world models often depend on simplifying assumptions.
- Clear mathematical communication should include the variable, equation, working, and contextual conclusion.
- Translating situations into equations is one of the most important applications of algebra.
- The complete process is understand → define → translate → solve → check → interpret.
Check Your Understanding
- What is mathematical modelling?
- Why should you define a variable before writing an equation?
- Translate into an equation: "A number increased by 9 is 25."
- Translate: "Five times a number is 45."
- Translate: "Seven less than twice a number is 19."
- A taxi charges $6 plus $4 per kilometer. The total fare is $34. Write and solve an equation to find the distance traveled.
- Five identical books plus a $10 delivery fee cost $85. Find the cost of each book.
- A student has $40 and saves $15 each week. How many weeks will it take to reach $145?
- A gym charges a $30 joining fee plus $12 per month. A customer has paid $102. How many months has the customer been a member?
- A car travels 240 km at 80 km/h. Use an equation to determine the travel time.
- A rectangle has width x and length x + 5. Its perimeter is 42 cm. Find its dimensions.
- Two supplementary angles are x and 2x. Find both angles.
- Two consecutive integers have a sum of 57. Find the integers.
- Company A charges $20 plus $4 per unit. Company B charges $32 plus $2 per unit. Determine when their costs are equal.
- An object has a mass of 30 kg and experiences a net force of 150 N. Use F = ma to determine its acceleration.
- A calculation gives 3.4 buses. Explain why 3.4 is not an appropriate final answer if every passenger must be transported and determine the likely practical answer.
- Explain the difference between solving an equation and interpreting its solution.
- Why should units be included in a final real-world answer?
- What should you do if an equation gives a mathematically correct answer that is impossible in the original context?
- Describe the complete process you would use to turn a real-world situation into an equation, solve it, and communicate the answer clearly.