Solving Linear Equations

Linear Equations

Lesson Objectives

By the end of this lesson, students should be able to solve linear equations algebraically and apply them in solving word problems.

Lesson Introduction

Solving linear equations is a fundamental skill in algebra that involves finding the value of an unknown variable. It's like playing a detective game, where you use clues and logical reasoning to uncover the hidden value. In this article, we'll explore different types of equations and the strategies used to solve them.

Linear Equations

Linear equations are equations where the highest power of the variable is 1. A linear equation is an equation that can be written in the form $Ax + B = C$ , where A, B, and C are constants and $x$ is the variable. They can be solved by isolating the variable on one side of the equation.

Examples

Example 1: Solve $2x + 5 = 13$ .

$2x = 13 - 5$ → $2x = 8$ → $x = \frac{8}{2} = 4$

Answer: $x = 4$

Example 2: The sum of the ages of a grandfather, father, and son is 130. The grandfather is twice as old as the father. The father is 10 years older than the son. Find the age of each.

Let the son's age be $x$ .

Then the father's age = $x + 10$ , and grandfather's age = $2(x + 10)$

Total: $x + (x + 10) + 2(x + 10) = 130$

$x + x + 10 + 2x + 20 = 130$ → $4x + 30 = 130$ → $4x = 100$ → $x = 25$

Answer: Son = 25, Father = 35, Grandfather = 70

Example 3: Solve $3(x - 2) = 2x + 4$ .

$3x - 6 = 2x + 4$ → $3x - 2x = 4 + 6$ → $x = 10$

Answer: $x = 10$

Example 4: Solve $\frac{2x - 1}{3} = 5$ .

$2x - 1 = 15$ → $2x = 16$ → $x = 8$

Answer: $x = 8$

Example 5: Solve $5x + 2 = 3x - 6$ .

$5x - 3x = -6 - 2$ → $2x = -8$ → $x = -4$

Answer: $x = -4$

Example 6: Solve $4(x + 3) = 2x + 18$ .

$4x + 12 = 2x + 18$ → $4x - 2x = 18 - 12$ → $2x = 6$ → $x = 3$

Answer: $x = 3$

Example 7: Solve $\frac{x + 2}{4} + \frac{x - 1}{2} = 5$ .

LCM = 4: $\frac{x + 2 + 2(x - 1)}{4} = 5$

$x + 2 + 2x - 2 = 20$ → $3x = 20$ → $x = \frac{20}{3}$

Answer: $x = \frac{20}{3}$

Example 8: Solve $0.5x + 1.5 = 4$ .

$0.5x = 4 - 1.5 = 2.5$ → $x = \frac{2.5}{0.5} = 5$

Answer: $x = 5$

Example 9: Solve $2(x - 1) = 2x - 2$ .

$2x - 2 = 2x - 2$ → All terms cancel, so: infinitely many solutions.

Answer: Infinite solutions

Example 10: Solve $3x + 5 = 3x + 9$ .

Subtract $3x$ both sides: $5 = 9$ → Contradiction.

Answer: No solution

Exercises

[WAEC] Solve $2x + 3 = 11$ . (Past Question)
Solve $5x - 7 = 18$ .
Solve $3(x - 4) = 2x + 1$ .
[NECO] Solve $\frac{2x + 1}{3} = 7$ . (Past Question)
Solve $4x + 3 = 3x + 10$ .
[JSCE] Solve $5x - 2 = 3x + 8$ . (Past Question)
Solve $6x + 4 = 2x + 20$ .
[JAMB] Solve $3x + 1 = x + 9$ . (Past Question)
Solve $\frac{x - 2}{5} = \frac{x + 1}{3}$ .
Solve $7x - 5 = 3x + 15$ .

Conclusion/Recap

Linear equations are equations of the form $Ax + B = C$ . Solving them involves simplifying expressions, using inverse operations, and isolating the variable. Always check your solution by substituting it back into the original equation. Understanding linear equations builds a foundation for solving more complex algebraic problems.