Algebra · Algebraic Expressions · Lesson 4

Substituting Values into Algebraic Expressions

An expression such as $1.5 u + 12$ is like a recipe that works for any number: put in the number of electricity units you used and out comes your bill. In this lesson you will replace letters by numbers — including negative numbers and fractions — and work out the value of expressions using the correct order of operations.

NaCCA B7.2.2.1.4 ⏱ about 45 min

By the end you will be able to

✔Substitute given values for letters in an algebraic expression.
✔Use the order of operations (BODMAS) correctly after substituting.
✔Substitute negative numbers safely using brackets.
✔Simplify an expression first, then substitute to evaluate it.
✔Use formulas such as perimeter, area and cost to solve real-life problems.

Key words

Substitute: Replace a letter by a given number.
Evaluate: Work out the value (a single number) of an expression.
Formula: A rule written with letters that connects quantities, e.g. Area $= l w$ .
BODMAS: The order of operations: Brackets, Of/Orders (powers), Division and Multiplication, Addition and Subtraction.

Curriculum reference

Content standard: B7.2.2.1 Simplify algebraic expressions involving the four basic operations and substituting values to evaluate algebraic expressions.

Indicator: Substitute values to evaluate algebraic expressions.

Starter

How much is the light bill?

Yaw's family in Koforidua buys prepaid electricity. Suppose the bill is worked out with the rule "GH₵1.50 for every unit used, plus a fixed charge of GH₵12", which we can write as $1.5 u + 12$ cedis for u units. Last month they used 100 units. This month they used 240. The rule stays the same; only the number for u changes. Putting the number in place of the letter is called substitution, and it is how every formula — from your school fees to a trotro fare — is used.

1. What substitution means

A letter in an expression is a box waiting for a number. To evaluate an expression we substitute: we take the letter out and put the given number in its place. Then we calculate. The secret is to remember the hidden signs: $3 a$ means $3 \times a$ , $ab$ means $a \times b$ and $\frac{x}{2}$ means $x \div 2$ .

1Write the expression down.
2Rewrite it with each letter replaced by its number, putting in the × signs (and brackets around negative numbers).
3Calculate using BODMAS: powers, then × and ÷, then + and −.
4Write the final value clearly.

▶ Each letter is replaced by its number. The hidden × signs appear, and BODMAS tells us to multiply before adding.

2. Order of operations and negative numbers

After substituting you have an ordinary calculation, so the usual order of operations applies. For example, if $c = 3$ then $2 c^{2} = 2 \times 3^{2} = 2 \times 9 = 18$ . The power belongs only to the $c$ , so we square 3 first and then multiply by 2 — not $6^{2} = 36$ .

Negative values need extra care. Put them in brackets when you substitute, so you do not lose the sign. Let $c = 3$ and $d = - 1$ :

Expression	Substitute	Value
$2 c - 5 d$	$2 (3) - 5 (- 1) = 6 + 5$	11
$d^{2}$	$(- 1)^{2} = (- 1) \times (- 1)$	1
$d^{2} + 3 d$	$1 + 3 (- 1) = 1 - 3$	−2
$c d$	$3 \times (- 1)$	−3

3. Simplify first, then substitute

When an expression is long, simplify it before substituting. You do less arithmetic and make fewer mistakes. With $a = 2$ and $b = 4$ :

$7 a + 5 b - 4 a + 2 b - 3 a$ : the a-terms give $7 - 4 - 3 = 0$ , the b-terms give $5 + 2 = 7$ , so the expression is $7 b = 7 \times 4 = 28$ .
$3 a \times 5 b = 15 ab = 15 \times 2 \times 4 = 120$ .
$\frac{12 ab}{16 a} = \frac{3 b}{4} = \frac{3 \times 4}{4} = 3$ .

4. Formulas in real life

A formula is an expression with a job to do. Substituting into formulas lets you find perimeters, areas, costs and bills. For a rectangle of length $l$ and width $w$ : perimeter $= 2 l + 2 w$ and area $= l w$ .

▶ A rectangle 8 m by 5 m: substitute l = 8 and w = 5 into area = lw to get 40 m².

Formula	Values	Working	Answer
Perimeter $= 2 l + 2 w$	$l = 8, w = 5$	$2 (8) + 2 (5) = 16 + 10$	26 m
Area $= l w$	$l = 8, w = 5$	$8 \times 5$	40 m²
Triangle area $= \frac{1}{2} bh$	$b = 8, h = 6$	$\frac{1}{2} \times 8 \times 6$	24 m²
Bill $= 1.5 u + 12$	$u = 100$	$1.5 \times 100 + 12$	GH₵162

Worked examples

Try each one yourself first, then open the solution and compare every step.

Example 1

A first substitution

Find the value of

3 a + 2 b

when

a = 2

and

b = 4

Show the solution

Step 1Substitute: $3 \times 2 + 2 \times 4$ .
Step 2Multiply first: $6 + 8$ .
Step 3Add: 14.

Answer: 14

Example 2

Simplify, then substitute

Simplify

3 a \times 5 b

and find its value when

a = 2

and

b = 4

Show the solution

Step 1Simplify: $3 a \times 5 b = 15 ab$ .
Step 2Substitute: $15 \times 2 \times 4 = 120$ .
Step 3Check: $3 a = 6$ , $5 b = 20$ , $6 \times 20 = 120$ ✓

Answer: $15 ab = 120$

Example 3

Negative values

c = 3

and

d = - 1

, find (a)

2 c - 5 d

(b)

c d^{2}

Show the solution

Step 1(a) $2 (3) - 5 (- 1) = 6 - (- 5) = 6 + 5 = 11$ .
Step 2(b) The power acts on $d$ only: $d^{2} = (- 1)^{2} = 1$ .
Step 3So $c d^{2} = 3 \times 1 = 3$ .

Answer: (a) 11 (b) 3

Example 4

A long expression

Simplify

7 a + 5 b - 4 a + 2 b - 3 a

and evaluate it when

a = 2

and

b = 4

Show the solution

Step 1a-terms: $7 a - 4 a - 3 a = 0$ .
Step 2b-terms: $5 b + 2 b = 7 b$ .
Step 3Substitute: $7 \times 4 = 28$ .
Step 4Check in the original: 14 + 20 − 8 + 8 − 6 = 28 ✓

Answer: $7 b = 28$

Example 5

Perimeter and area

A rectangular garden has length

l = 8

m and width

w = 5

m. Use

P = 2 l + 2 w

and

A = l w

to find its perimeter and area.

Show the solution

Step 1 $P = 2 (8) + 2 (5) = 16 + 10 = 26$ m.
Step 2 $A = 8 \times 5 = 40$ m².

Answer: Perimeter 26 m, area 40 m²

Summary

To substitute, replace each letter by its value and put back the hidden × signs: $3 a$ with $a = 2$ is $3 \times 2 = 6$ .
After substituting, follow BODMAS: powers first, then × and ÷, then + and −.
Put negative values in brackets: $(- 1)^{2} = 1$ and $5 - 3 (- 1) = 8$ .
Simplify long expressions first, then substitute — less work, fewer mistakes.
Formulas such as $P = 2 l + 2 w$ and $A = l w$ are used by substitution.

How much is the light bill?

1. What substitution means

2. Order of operations and negative numbers

3. Simplify first, then substitute

4. Formulas in real life

Worked examples

A first substitution

Simplify, then substitute

Negative values

A long expression

Perimeter and area

Summary

Practice: 15 questions