Introduction

An algebraic equation is a mathematical statement showing that two expressions have the same value.

For example:

3x + 5 = 20

The equal sign tells us that the expression on the left has exactly the same value as the expression on the right.

Solving an equation means finding the value of the variable that makes this statement true.

Algebraic equations are used throughout mathematics and in real-world situations involving money, distance, time, measurement, science, and engineering.

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Parts of an Algebraic Equation

Consider:

4x + 7 = 23

Several important algebraic terms appear in this equation.

Variable

A variable is a symbol, usually a letter, that represents an unknown or changing value.

In our example:

x​

is the variable.

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Coefficient

A coefficient is the number multiplying a variable.

In:

4x

the coefficient is:

4​

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Constant

A constant is a number that does not contain a variable.

In:

4x + 7 = 23

the numbers 7 and 23 are constants.

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Terms

A term is a number, variable, or product of numbers and variables separated by addition or subtraction.

For example:

5x + 3y − 8

contains three terms:

5x, 3y, −8

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Understanding Equality

The symbol:

=

means "has the same value as."

For example:

3 + 4 = 7

Both sides have the same value.

Similarly:

x + 4 = 10

means that the value of x + 4 must equal 10.

The value that makes this statement true is:

x = 6

because:

6 + 4 = 10

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The Balance Model

A useful way to think about equations is as a balanced scale.

If:

x + 3 = 8

then both sides of the equation have equal value.

If we subtract 3 from one side, we must also subtract 3 from the other:

x + 3 − 3 = 8 − 3

Therefore:

x = 5

Whatever operation we perform on one side of an equation must also be performed on the other side.

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Inverse Operations

To solve equations, we often use inverse operations.

Inverse operations undo each other.

For example:

x + 7 = 12

Since 7 has been added, subtract 7:

x + 7 − 7 = 12 − 7

Therefore:

x = 5​

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Solving One-Step Equations

A one-step equation requires one inverse operation to isolate the variable.

Example 1: Addition

Solve:

x + 9 = 15

Subtract 9 from both sides:

x + 9 − 9 = 15 − 9

 

x=6​

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Example 2: Subtraction

Solve:

x − 4 = 11

Add 4 to both sides:

x − 4 + 4 = 11 + 4

 

x = 15​

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Example 3: Multiplication

Solve:

5x = 30

Divide both sides by 5:

\( \frac{5x}{5} = \frac{30}{5} \)​

 

x = 6​

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Example 4: Division

Solve:

\( \frac{x}{4} = 7 \)

Multiply both sides by 4:

\( 4( \frac{x}{4}) = 4(7) \)

 

x = 28​

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Solving Two-Step Equations

A two-step equation requires two inverse operations.

Consider:

3x + 5 = 20

Our goal is to isolate x.

Step 1: Remove the constant

Subtract 5 from both sides:

3x + 5 − 5 = 20 − 5

 

 

Step 2: Remove the coefficient

Divide both sides by 3:

​

Therefore:

x = 5

​

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Another Two-Step Example

Solve:

4x − 7 = 21

First, add 7 to both sides:

4x = 28

Then divide both sides by 4:

x = 7

​

A useful strategy is to undo operations in reverse order.

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Checking a Solution

After solving an equation, we can verify our answer using substitution.

Suppose:

2x + 4 = 16

We solve and obtain:

x = 6

Now substitute 6 for x:

2(6) + 4 = 16

 

 

16 = 16

The statement is true, so:

x = 6​

is the correct solution.

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Is a Value a Solution?

Sometimes we are given a value and asked whether it solves an equation.

Example

Is x = 4 a solution to:

3x + 2 = 14?

Substitute x=4:

3(4) + 2 = 14

 

 

14 = 14

The statement is true.

Therefore:

x = 4​

is a solution.

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Example of an Incorrect Solution

Is x = 5 a solution to:

2x + 3 = 15?

Substitute:

2(5) + 3 = 15

 

 

Therefore:

x = 5​

is not a solution.

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A Strategy for Solving Equations

When solving an equation:

1. Identify the variable.
2. Determine which operations are being applied to it.
3. Use inverse operations to undo them.
4. Perform the same operation on both sides.
5. Continue until the variable is isolated.
6. Substitute your answer back into the original equation to check it.

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Equations in Real Life

Algebraic equations allow us to represent unknown quantities in real situations.

Example

A taxi charges a fixed fee of $4 plus $3 per kilometre.

A journey costs $25.

How far did the taxi travel?

Let:

x = distance travelled

The equation is:

3x + 4 = 25

Subtract 4:

3x = 21

Divide by 3:

x = 7​

The taxi travelled 7 km.

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Translating Words into Equations

Understanding common mathematical phrases helps us create equations.

Turning words into equations allows us to solve many practical problems.

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Common Mistakes

Mistake 1: Changing only one side

Incorrect:

x + 5 = 12

 

x + 5 − 5 = 12

You must perform the same operation on both sides.

Correct:

x + 5 − 5 = 12 − 5

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Mistake 2: Using the wrong inverse operation

To undo:

x + 8

use:

− 8

not + 8.

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Mistake 3: Forgetting to check the solution

Substitution provides a quick way to detect arithmetic errors.

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Worked Examples

Example 1

Identify the variable and coefficient in:

7x + 4 = 25

Answer:

• Variable: x
• Coefficient: 7

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Example 2

Solve:

x − 9 = 14

Add 9:

x = 23

Answer:

x = 23​

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Example 3

Solve:

6x = 42

Divide by 6:

x = 7​

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Example 4

Solve:

5x + 3 = 28

Subtract 3:

5x = 25

Divide by 5:

x = 5​

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Example 5

Solve and check:

3x − 2 = 19

Add 2:

3x = 21

Divide by 3:

x = 7

Check:

3(7) − 2 = 19

 

 

19 = 19

Therefore:

x = 7​

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Did You Know?

The word algebra comes from the Arabic word al-jabr. It appeared in the title of an influential mathematics book written around 820 CE by the mathematician Muhammad ibn Musa al-Khwarizmi.

His work helped develop systematic methods for solving equations and played an important role in the development of modern algebra.

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Key Terms

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Key Takeaways

• An equation states that two mathematical expressions have equal value.
• Equations contain important parts such as variables, coefficients, constants, and terms.
• The equal sign means that the expressions on both sides have the same value.
• To maintain equality, whatever operation is performed on one side must also be performed on the other.
• Inverse operations allow us to undo operations and isolate the variable.
• One-step equations require one inverse operation, while two-step equations require two.
• A solution can be verified by substituting it back into the original equation.
• Algebraic equations allow us to represent and solve many mathematical and real-world problems.