Linear Equations

Grade 10 Math Lesson - Linear Equations

Lesson Objectives

Define and understand the concept of linear equations.
Solve simple and complex linear equations in one variable.
Apply the distributive property and like terms in solving equations.
Translate and solve real-life word problems involving linear equations.

Lesson Introduction

Linear equations are the foundation of algebra. They appear in everyday situations like budgeting, measuring distance, or determining the price of goods. Understanding how to form and solve these equations helps us solve real-world problems using mathematical logic.

Core Lesson Content

Definition: A linear equation is an equation in which the highest power of the variable is 1.

General Form: $ax + b = 0$ where $a \neq 0$

To solve a linear equation:

Isolate the variable.
Use inverse operations (addition, subtraction, multiplication, division).
Always perform the same operation on both sides of the equation.

Worked Example

Example 1: Solve $x + 5 = 12$

Subtract 5 from both sides:

x = 12 - 5

x = 7

Example 2: Solve $3x = 21$

Divide both sides by 3:

x = 21 \div 3

x = 7

Example 3: Solve $2x - 4 = 10$

Add 4 to both sides:

2x = 14

Divide both sides by 2:

x = 14 \div 2 = 7

Example 4: Solve $5x + 3 = 3x + 15$

Move like terms:

5x - 3x = 15 - 3

2x = 12

x = 6

Example 5: Solve $\frac{x}{2} + 3 = 7$

Subtract 3:

\frac{x}{2} = 4

Multiply both sides by 2:

x = 4 \times 2 = 8

Example 6: Solve $2(x - 1) = 10$

Expand brackets:

2x - 2 = 10

Add 2:

2x = 12

Divide by 2:

x = 6

Example 7 (Word Problem): If twice a number minus 3 equals 7, find the number.

Let the number be $x$ :

2x - 3 = 7

2x = 7 + 3 = 10

x = 10 \div 2 = 5

Example 8 (Word Problem): A number added to its half gives 18. What is the number?

Let the number be $x$ :

x + \frac{x}{2} = 18

\frac{3x}{2} = 18

3x = 36

x = 12

Exercises

Solve: $x + 9 = 20$
Solve: $4x - 5 = 15$
Solve: $\frac{x}{3} + 4 = 9$
Solve: $7x + 2 = 3x + 18$
[WAEC] Solve for $x$ : $5(x - 2) = 3x + 6$ (Past Question)
Solve: $3(x + 4) = 21$
[NECO] If $2x + 5 = x + 12$ , find $x$ . (Past Question)
[JAMB] A number when doubled and reduced by 6 gives 10. Find the number. (Past Question)
Solve: $6x - (2x + 4) = 8$
[WAEC] Solve: $\frac{2x + 1}{3} = 5$ (Past Question)

Conclusion/Recap

In this lesson, we learned how to solve various types of linear equations, including those embedded in real-life problems. Mastering this topic prepares you for advanced algebra topics like inequalities and simultaneous equations. Up next: Solving Linear Inequalities.