Simple Equations

Grade 12 Math - Solving Linear Equations

Lesson Objectives

Solve linear equations involving one unknown.
Apply linear equations to solve word problems.
Interpret real-life scenarios into solvable algebraic equations.

Lesson Introduction

Linear equations are equations of the first degree and involve variables raised only to the power of one. In this lesson, we will solve linear equations algebraically and apply them to real-world word problems. This skill is foundational for algebra, finance, and logical reasoning.

Core Lesson Content

Worked Example

Solving Linear Equations

A linear equation has the general form: $ax + b = 0$ , where a and b are constants.

Example 1: Solve

3x - 5 = 10

Add 5 to both sides:

3x = 15

Divide by 3:

x = 5

Example 2: Solve

2x + 4 = 3x - 6

Bring all terms to one side:

2x - 3x = -6 - 4

-x = -10 \Rightarrow x = 10

Example 3: Solve

\frac{2x + 1}{3} = \frac{x - 2}{2}

Cross-multiply:

2(2x + 1) = 3(x - 2)

4x + 2 = 3x - 6

x = -8

Solving Word Problems with Linear Equations

Steps:

Define the unknown(s).
Translate the problem into an equation.
Solve the equation.
Interpret the result in context.

Example 4: The sum of two consecutive odd numbers is 36. Find the numbers.
Let the first number be

x

, the next is

x + 2

x + (x + 2) = 36 \Rightarrow 2x + 2 = 36 \Rightarrow 2x = 34 \Rightarrow x = 17

So the numbers are 17 and 19.

Example 5: A father is four times as old as his son. In 5 years, he will be three times as old as his son. What are their current ages?
Let son's age be

x

, father's age =

4x

In 5 years:

4x + 5 = 3(x + 5)

4x + 5 = 3x + 15 \Rightarrow x = 10

Son is 10, father is 40

Example 6: A trader bought a bag of rice for ₦x and sold it for ₦14,000, making a profit of ₦2,000. Find the cost price.

x + 2000 = 14000 \Rightarrow x = 12000

The cost price is ₦12,000.

Exercises

Solve $5x + 3 = 2x + 18$
[WAEC] Solve $4(2x - 1) = 3(x + 5)$ [Past Question]
A number is 5 less than three times another number. If their sum is 31, find the numbers.
Solve $\frac{3x - 4}{2} = \frac{x + 5}{3}$
[NECO] John is 6 years older than James. In 4 years, the sum of their ages will be 40. Find their current ages. [Past Question]
The length of a rectangle is twice its width. If the perimeter is 60 cm, find its dimensions.
Find $x$ such that $2x + 7 = 3x - 5$
[WAEC] A father is twice as old as his daughter. Five years ago, he was three times her age. How old are they now? [Past Question]
[NECO] A piece of rope is cut into two parts. One part is 4 m longer than the other. If the total length is 28 m, find the length of each part. [Past Question]
Solve $\frac{2x + 3}{4} - \frac{x - 2}{2} = 3$

Conclusion/Recap

Linear equations allow us to model and solve real-life problems mathematically. Understanding how to interpret word problems and translate them into equations is key to applying math in practical contexts. Mastery of linear equations is a foundation for algebra, geometry, and future topics such as systems of equations and functions.