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.

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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.

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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

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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

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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.

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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.

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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

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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.

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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.

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Operations

Equations can contain mathematical operations such as:

• addition (+)
• subtraction (−)
• multiplication (×)
• division (÷)
• powers

For example:

3x + 5 = 20

contains multiplication and addition.

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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.

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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.

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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.

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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

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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

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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

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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

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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

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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

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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

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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.

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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

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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.

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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.

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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.

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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.

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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

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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.

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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.

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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.

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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.

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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.

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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

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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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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

1. What is an equation?
2. What does the equals sign mean?
3. Explain the difference between an expression and an equation.
4. Identify the variable in 5x + 7 = 32.
5. Identify the coefficient of x in 5x + 7 = 32.
6. Identify the constants in 5x + 7 = 32.
7. What is the left-hand side of 3y − 4 = 17?
8. What is the right-hand side?
9. Explain why an equation can be compared with a balanced scale.
10. If 7 is added to one side of an equation, what must be done to the other side to preserve equality?
11. Is x = 4 a solution to 2x + 3 = 11? Verify by substitution.
12. Is x = 5 a solution to 3x − 2 = 10? Explain.
13. Verify whether x = −2 solves 4x + 3 = −5.
14. Verify whether y = 6 solves 2y − 5 = y + 1.
15. Explain what substitution means.
16. Why is it useful to substitute a solution back into the original equation?
17. Give an example of an equation with no solution.
18. Give an example of an equation that is true for every value of x.
19. Write an equation for: "Three times a number plus 4 equals 19."
20. Explain in your own words why maintaining equality is the central idea behind solving equations.