YEAR 8 · ALGEBRAIC EXPRESSIONS · 13 MIN

Substitution with signed/rational values

Evaluate expressions using signed and fractional values, with brackets to preserve negative inputs.

In this lesson

Replace every variable by its supplied value
Bracket negative substitutions
Calculate powers and products before sums

Before you start

Order of operations
Signed and fraction arithmetic

01 / LEARN

Understand the idea.

Replace the variable, not the operation

Substitution replaces each variable by its given value while keeping the original operations. The expression $3 x + 2$ at $x = - 4$ becomes $3 (- 4) + 2$ . The coefficient still multiplies the input; it is not joined to its digits.

Put a negative input in brackets so its sign belongs to the whole value. Replace every occurrence of each variable, including fractional inputs. For example, $6 b$ at $b = \frac{1}{2}$ becomes $6 \times \frac{1}{2}$ , which equals $3$ .

Follow the operation order

After substitution, use the normal order of operations: brackets and powers, multiplication and division, then addition and subtraction. At $x = - 3$ , the expression $x^{2}$ becomes ${(- 3)}^{2} = 9$ , whereas $- x^{2}$ becomes $- 9$ .

For fractions, cancel common factors in a product before multiplying if this makes the arithmetic easier. Keep exact values until the calculation is complete, and check whether the sign and approximate size of the result fit the substituted numbers.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A negative input

Evaluate $4 x - 7$ for $x = - 2$ .

Worked solution

1
Replace x by the whole negative input.
$4 x - 7 = 4 (- 2) - 7$
Brackets make clear that the coefficient multiplies negative two.
2
Complete multiplication first.
$4 (- 2) - 7 = - 8 - 7$
Multiplication takes priority over subtraction.
3
Subtract the remaining seven.
$- 8 - 7 = - 15$
Subtracting seven moves seven units further below zero.

Answer

$- 15$

EXAMPLE 02

Two rational inputs

Evaluate $3 a + 4 b$ for $a = - 2, b = \frac{3}{4}$ .

Worked solution

1
Replace both variables.
$3 a + 4 b = 3 (- 2) + 4 (\frac{3}{4})$
Both given values must be used in their matching terms.
2
Evaluate each product.
$\begin{aligned} 3 (- 2) & = - 6 \\ 4 (\frac{3}{4}) & = 3 \end{aligned}$
A positive times a negative is negative, and four quarters make one whole.
3
Add the signed results.
$- 6 + 3 = - 3$
Adding three to negative six moves three units towards zero.

Answer

$- 3$

03 / YOUR TURN

Try it yourself.

Check your understanding

Evaluate $2 x + 5$ for $x = - 3$ .

Show answer & reasoning

$- 1$

Substitute before calculating: $2 (- 3) + 5 = - 6 + 5 = - 1$ . Multiply first, then add the positive five to negative six.

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Evaluate $5 x + 1$ for $x = - 2$ .
2 marks
Evaluate $a^{2} + 2$ for $a = - 4$ .
2 marks
Evaluate $6 b - 5$ for $b = \frac{1}{2}$ .
2 marks
Evaluate $2 p - 3 q$ for $p = - 1, q = 2$ .
2 marks
Evaluate $4 m + 2 n$ for $m = \frac{3}{4}, n = - 2$ .
2 marks
Evaluate $3 (x - 2)$ for $x = - 1$ .
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Evaluate $5 x + 1$ for $x = - 2$ .
2 marks
$5 (- 2) + 1 = - 10 + 1 = - 9$ .
$- 9$
Evaluate $a^{2} + 2$ for $a = - 4$ .
2 marks
${(- 4)}^{2} + 2 = 16 + 2 = 18$ .
$18$
Evaluate $6 b - 5$ for $b = \frac{1}{2}$ .
2 marks
$6 \times \frac{1}{2} - 5 = 3 - 5 = - 2$ .
$- 2$
Evaluate $2 p - 3 q$ for $p = - 1, q = 2$ .
2 marks
$2 (- 1) - 3 (2) = - 2 - 6 = - 8$ .
$- 8$
Evaluate $4 m + 2 n$ for $m = \frac{3}{4}, n = - 2$ .
2 marks
$4 \times \frac{3}{4} + 2 (- 2) = 3 - 4 = - 1$ .
$- 1$
Evaluate $3 (x - 2)$ for $x = - 1$ .
2 marks
The bracket is $- 1 - 2 = - 3$ .
Multiply $3 (- 3) = - 9$ .
$- 9$

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Curriculum reference & lesson scope

Australian Curriculum v9 Mathematics with SA curriculum context

Combining like terms; substitution with signed/rational values; multiplying/dividing algebraic terms; expanding a single bracket; factorising common factors; equivalent linear expressions; rearranging simple formulas.

Official curriculum source ↗

In this lesson

Understand the idea.

Replace the variable, not the operation

Follow the operation order

See the method in action.

A negative input

Replace x by the whole negative input.

Complete multiplication first.

Subtract the remaining seven.

Two rational inputs

Replace both variables.

Evaluate each product.

Add the signed results.

Try it yourself.

Check your understanding

Keep practising