Introduction to Algebra

Lesson Objectives

By the end of this lesson, students should be able to:

Define algebra and understand its basic concepts
Differentiate between types of algebra
Simplify algebraic expressions
Solve simple algebraic equations

Lesson Introduction

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It involves expressions, equations, variables, and constants. Understanding algebra is crucial because it forms the foundation of most mathematical concepts you'll encounter later in school and real life.

What is Algebra?

Algebra uses letters and symbols to represent numbers and quantities in equations and formulas. These symbols represent unknown values and help us form general rules about relationships between quantities.

Key Concepts in Algebra

Variables: Symbols like $x$ , $y$ , or $z$ used to represent unknown values.
Constants: Fixed numerical values like 2, -5, or 3.14.
Expressions: Mathematical phrases like $2x + 5$ that can be simplified but not solved.
Equations: Statements with an equal sign, like $2x + 3 = 9$ , which can be solved to find the value of the variable.

Types of Algebra

Elementary Algebra: Basic algebra involving simple equations and expressions.
Abstract Algebra: Focuses on algebraic structures like groups, rings, and fields (not covered in this level).
Linear Algebra: Deals with vectors, matrices, and linear transformations.

Simplifying Algebraic Expressions

This involves combining like terms and using arithmetic operations to reduce expressions.

Example 1: Simplify $3x + 4x - 2 + 5$

$3x + 4x = 7x$ ; $-2 + 5 = 3$

Answer: $7x + 3$

Example 2: Simplify $2(x + 3) + 4(x - 2)$

$2x + 6 + 4x - 8 = 6x - 2$

Answer: $6x - 2$

Solving Linear Equations

To solve an equation, isolate the variable on one side using inverse operations.

Example 3: Solve $2x + 3 = 9$

Subtract 3 from both sides: $2x = 6$

Divide both sides by 2: $x = 3$

Answer: $x = 3$

Example 4: Solve $5x - 4 = 3x + 6$

Move variables to one side: $5x - 3x = 6 + 4$

$2x = 10$ → $x = 5$

Answer: $x = 5$

Using Algebra in Word Problems

Example 5: The sum of three consecutive numbers is 72. Find the numbers.

Let the numbers be $x, x+1, x+2$

$x + x + 1 + x + 2 = 72$ → $3x + 3 = 72$

$3x = 69$ → $x = 23$

Answer: $23, 24, 25$

Example 6: John is 4 years older than Peter. If the sum of their ages is 32, how old is each?

Let Peter's age be $x$ , then John's is $x + 4$

$x + x + 4 = 32$ → $2x = 28$ → $x = 14$

Answer: Peter = 14, John = 18

Transposition and Solving with Fractions

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

Subtract 3: $\frac{x}{2} = 4$ → Multiply both sides by 2: $x = 8$

Answer: $x = 8$

Example 8: Solve $\frac{2x - 1}{3} = 5$

Multiply both sides by 3: $2x - 1 = 15$

Add 1: $2x = 16$ → $x = 8$

Answer: $x = 8$

Solving Equations with Brackets

Example 9: Solve $2(x + 5) = 18$

Expand: $2x + 10 = 18$ → $2x = 8$ → $x = 4$

Answer: $x = 4$

Example 10: Solve $3(x - 2) + 4 = 19$

Expand: $3x - 6 + 4 = 19$ → $3x - 2 = 19$

$3x = 21$ → $x = 7$

Answer: $x = 7$

Exercises

[WAEC] Solve $3x + 4 = 16$ .
(Past Question)
Simplify $2x + 3x - 5 + 2$ .
[NECO] Solve $4(x - 1) = 12$ .
(Past Question)
Find $x$ if $x + \frac{x}{2} = 6$ .
[JAMB] Solve $2x - 3 = x + 5$ .
(Past Question)
Simplify $5(x + 2) - 3x$ .
[NABTEB] Solve $\frac{3x + 1}{2} = 7$ .
(Past Question)
Evaluate $2(x + 4) + 3(x - 2)$ .
If $x + y = 10$ and $x - y = 2$ , find the values of $x$ and $y$ .
Solve $\frac{x - 2}{4} + 1 = 3$ .

Conclusion/Recap

In this lesson, we explored the foundations of algebra—covering variables, expressions, and equations. We also examined how to simplify and solve equations, including those with brackets and fractions. With regular practice, these skills will help you tackle more advanced problems in future topics like quadratic equations, functions, and inequalities.