Substitution with signed/rational values | Year 8

01 / LEARN

The idea, step by step

Replace the variable, not the operation

Substitution evaluates an expression at given values. In $3 x + 2$ with $x = - 4$ , replace $x$ by $(- 4)$ to get $3 (- 4) + 2$ . The coefficient still means multiplication. A negative input belongs inside brackets.

With multiple variables, list their values and replace every occurrence. Fractional inputs are numbers too: $6 b$ at $b = \frac{1}{2}$ means $6 \times \frac{1}{2} = 3$ .

Follow the operation order

After substituting, complete brackets and powers, then multiplication and division, then addition and subtraction. With $x = - 3$ , $x^{2} = {(- 3)}^{2} = 9$ . This is different from $- x^{2}$ , the negative of that square.

Check the final sign and size. Keep an exact fraction when convenient; converting every fraction to a rounded decimal can create an avoidable error. A small value table can help check repeated substitutions.

A common trap

Replacing $x^{2}$ by $- 9$ when $x = - 3$ .

Instead: The whole input is squared: $(- 3) (- 3) = 9$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

A negative input

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

Substitute $4 (- 2) - 7$ .
Multiply first: $- 8 - 7$ .
Subtract seven from negative eight.

- 15

Example 2

Two rational inputs

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

Substitute $3 (- 2) + 4 (\frac{3}{4})$ .
The products are $- 6$ and $3$ .
Add $- 6 + 3 = - 3$ .

- 3

Check your understanding

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

Show answer

$- 1$

$2 (- 3) + 5 = - 6 + 5 = - 1$ .

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF 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 $3 (- 3) = - 9$ .
$- 9$

Checking saved progress…

Continue learning

Multiplying/dividing algebraic terms

Next lesson

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 ↗

Back to Algebraic expressions