Solving One-Step and Multi-Step 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.