Substitution

Substitution involves replacing a variable in an equation or formula with a given number or expression. This technique helps us to solve equations or evaluate expressions.

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For example, if we have an equation $3x + 4$ , and we are told that $x = 2$ , we can substitute 2 in place of $x$ to get $3 \times 2 + 4 = 10$ .

The Basics of Substitution

Single variable substitution

In its simplest form, single variable substitution involves replacing a variable in an equation or expression with a specific number or another expression.

Let’s look at a numerical substitution.

Example 1:

Consider the equation $5 \times x - 3$ , and we’re told $x = 4$ .

1. Substitute 4 for x: $5 \times 4 - 3$

2. Simplify to $20 - 3$

$= 17$

Multiple variable substitution

Sometimes an expression or equation will contain more than one variable. You may need to substitute values for each variable to evaluate or simplify it.

Example 2:

Consider $2 \times x + 3 \times y$ , and we’re told $x = 3$ and $y = 2$ .

1. Substitute 3 for x and 2 for y: $2 \times 3 + 3 \times 2$

2. Simplify to $6 + 6$

$= 12$

Substitution in formulas

Substitution isn’t limited to simple equations or expressions, it’s also important when working with formulas.

Example 1: Area of a Rectangle

For $A = l \times w$ , where l is the length and w is the width:

1. If $l = 7$ units and $w = 4$ units, substitute these into $A = 7 \times 4$

2. Simplify to $A = 28$ square units

Example 2: Distance Formula

The formula $d = v \times t$ calculates distance d based on speed v and time t.

1. If $v = 50 \ \text{km/h}$ and $t = 2$ hours, substitute into $d = 50 \times 2$

2. Simplify to $d = 100 \ \text{km}$

Here is a sample table that includes different expressions and the variables to be substituted:

Expression	Variable(s) to Substitute	Substituted Expression	Result
$2x + 3$	$x = 5$	$2 \times 5 + 3$	$13$
$4x - 5$	$x = -3$	$4 \times (-3) - 5$	$-17$
$3x + 4y$	$x = 2, y = 3$	$3 \times 2 + 4 \times 3$	$18$
$A = l \times w$	$l = 6, w = 5$	$6 \times 5$	$30$