Substitution in algebraic expressions is a critical skill that enables students to replace variables with given numerical values. This process simplifies expressions or equations, making them easier to evaluate or solve. Mastery of substitution is essential for success in mathematics and its applications.

What is Substitution?

Substitution involves replacing algebraic variables with specific numbers. It's a straightforward yet powerful technique to evaluate expressions and solve equations.

Image courtesy of All About Circuits

Key Concepts

• Variables as placeholders: They represent unknown or generalized numbers.
• Numerical substitution: Inserting numbers for variables to simplify expressions.
• Order of operations: Adhering to BIDMAS (Brackets, Indices, Division and Multiplication, Addition and Subtraction) when performing substitutions.

Substituting Values into Expressions

To substitute effectively:

1. Identify the variable(s) in the expression.

2. Determine the numerical value for each variable.

3. Substitute and simplify the expression according to the order of operations.

Example 1: Simple Substitution

Given the expression $3x + 4$ and $x = 2$, substitute and simplify:

1. Substitute $x$ with $2$: $3(2) + 4$

2. Simplify: $6 + 4 = 10$

Final answer: $10$

Example 2: Multiple Variables

For $2x + 3y$ with $x = 3$ and $y = 2$:

1. Substitute $x = 3$ and $y = 2$: $2(3) + 3(2)$

2. Simplify: $6 + 6 = 12$

Final answer: $12$

Substitution in Formulas

Substitution also applies to formulas in various mathematical and real-world contexts.

Example 3: Applying Formulas

To find the area $A$ of a rectangle where $l = 5$ cm and $w = 3$ cm:

1. Formula: $A = l \times w$

2. Substitute $l = 5$ and $w = 3$: $5 \times 3$

3. Simplify: $15$

Final answer: $15 \text{ cm }^2$

Practical Questions

Question 1

Given $4x - 7$ and $x = 3$, find the value.

1. Substitute: $4(3) - 7$

2. Simplify: $12 - 7 = 5$

Final answer: $5$

Question 2

For a square with perimeter $P = 20$ cm and $P = 4s$, find $s$.

1. Rearrange: $s = \frac{P}{4}$

2. Substitute: $s = \frac{20}{4}$

3. Simplify: $s = 5$

Final answer: $5$ cm

Question 3

If $y = 2x + 5$ and $x = 4$, find $y$.

1. Substitute: $y = 2(4) + 5$

2. Simplify: $y = 8 + 5 = 13$

Question 4

Calculate the value of $7x^2 - 4x + 3$ for $x = -1$.

1. Substitute: $7(-1)^2 - 4(-1) + 3$

2. Simplify: $7(1) + 4 + 3 = 14$