Algebraic Expressions and Formulae

Grade 10 Mathematics - Algebraic Expressions and Formulae

Lesson Objectives

Understand and simplify algebraic expressions.
Apply rules of addition, subtraction, and multiplication of algebraic terms.
Substitute given values into algebraic formulae correctly.
Solve real-world problems involving algebraic expressions and formulae.

Lesson Introduction

Algebra is often called the "language of mathematics." In everyday life, formulas are used to calculate speed, area, volume, and other important quantities. Mastering the simplification of expressions and correct substitution into formulae will make solving real-life problems much easier!

Core Lesson Content

What are Algebraic Expressions?

An algebraic expression is a mathematical phrase that includes numbers, variables, and operations like addition, subtraction, multiplication, and division. Examples include \( 2x + 5 \), \( 3a^{2} - 4a + 7 \).

Simplifying Expressions:

Simplifying means combining like terms (terms with the same variable part) and performing arithmetic operations.

Substituting into Formulae:

Substitution involves replacing variables with given values and calculating the result. Example: if \( A = l \times w \), then substituting \( l = 5 \), \( w = 3 \) gives \( A = 15 \).

Worked Examples

Example 1: Simplify \( 3x + 5x - 2x \)

Answer:

Combine like terms: \( (3 + 5 - 2)x = 6x \)

Example 2: Simplify \( 4a - 2b + 3a + b \)

Answer:

Group like terms: \( (4a + 3a) + (-2b + b) = 7a - b \)

Example 3: Expand and simplify \( 2(x + 3) + 4(x - 5) \)

Answer:

Expand: \( 2x + 6 + 4x - 20 \) Combine like terms: \( (2x + 4x) + (6 - 20) = 6x - 14 \)

Example 4: Simplify \( 5(2a - 3b) - 2(3a + 4b) \)

Answer:

Expand: \( 10a - 15b - 6a - 8b \) Combine like terms: \( (10a - 6a) + (-15b - 8b) = 4a - 23b \)

Example 5: If \( y = 2x + 5 \), find \( y \) when \( x = 3 \)

Answer:

Substitute: \( y = 2(3) + 5 = 6 + 5 = 11 \)

Example 6: Find the value of \( A = l \times w \) when \( l = 7 \) and \( w = 4 \)

Answer:

Substitute: \( A = 7 \times 4 = 28 \)

Example 7: If \( V = l \times b \times h \), find \( V \) when \( l = 5 \), \( b = 3 \), and \( h = 2 \)

Answer:

Substitute: \( V = 5 \times 3 \times 2 = 30 \)

Example 8: Simplify \( 6x^{2} - 2x + 3x^{2} + 5x \)

Answer:

Group like terms: \( (6x^{2} + 3x^{2}) + (-2x + 5x) = 9x^{2} + 3x \)

Example 9: Find \( p \) if \( p = 2a - 3b \), \( a = 4 \), \( b = 2 \)

Answer:

Substitute: \( p = 2(4) - 3(2) = 8 - 6 = 2 \)

Example 10: Simplify \( (2x - 3)(x + 5) \)

Answer:

Expand: \( 2x^{2} + 10x - 3x - 15 \) Combine like terms: \( 2x^{2} + 7x - 15 \)

Exercises

Simplify: \( 5x + 7x - 3x \)
Expand and simplify: \( 3(a + 4) - 2(2a - 1) \)
Find \( y \) if \( y = 4x - 7 \) when \( x = 5 \)
Expand: \( (x - 2)(x + 6) \)
Simplify: \( 2p - 3q + 5p + 2q \)
[WAEC] If \( A = \pi r^{2} \), find \( A \) when \( r = 7 \). (Past Question)
[NECO] Simplify: \( (3x - 2)(x + 5) \). (Past Question)
Find the volume \( V \) of a box using \( V = l \times b \times h \), if \( l = 8 \), \( b = 5 \), \( h = 2 \)
Simplify: \( 7a - 3b + 2a + 5b \)
[JAMB] If \( P = 2(l + w) \), find \( P \) when \( l = 6 \) and \( w = 4 \). (Past Question)

Conclusion/Recap

Today, you learned how to simplify algebraic expressions by combining like terms and how to substitute values correctly into formulae. These skills are foundational for solving equations and modeling real-life problems. Next lesson, we will move into solving simple and complex algebraic equations!