Evaluating Algebraic Expressions
Learn to find the value of expressions by substitution
1. What is Substitution?
The Big Idea
Substitution means replacing a letter with a number to find the value of an expression.
When we know the value of the letter, say y = 5, we can put 5 in place of y in the expression!
Example: The Magic of Substitution
If y = 5, find the value of y + 3.
Replace y with 5, then add 3
y + 3
= 5 + 3 (substitute y = 5)
= 8
2. Evaluating Addition and Subtraction
Example 1: The Stick and the Rod
A rod is y cm longer than a stick. The stick is 20 cm long.
(a) Express the length of the rod in terms of y.
Length of rod = (20 + y) cm
(b) Find the length of the rod when y = 9.
Length of rod = (20 + y) cm
= (20 + 9) cm
= 29 cm
Example 2: Hassan's Age Table
Hassan is 12 years old now. How old is he at different times?
| Time | Expression | If p = 3, q = 5, r = 5 |
|---|---|---|
| now | 12 | 12 |
| in 3 years' time | 12 + 3 | 15 |
| in p years' time | 12 + p | 12 + 3 = 15 |
| in q years' time | 12 + q | 12 + 5 = 17 |
| 5 years ago | 12 - 5 | 7 |
| r years ago | 12 - r | 12 - 5 = 7 |
Practice: Substitution
If n = 7, find the value of 25 - n.
ShowHide solution
25 - n
= 25 - 7 (substitute n = 7)
= 18
3. Evaluating Multiplication and Division
Remember: 4x means 4 × x. So when x = 5, we calculate 4 × 5 = 20.
Example: Cherries in Three Boxes
Box A has x cherries. Box B has 3 times as many cherries as Box A. Box C has 9 cherries.
(a) Express the total number of cherries in terms of x.
Total = x + 3x + 9
= 4x + 9
(b) Find the total number of cherries when x = 15.
Total = 4x + 9
= 4 × 15 + 9
= 60 + 9
= 69 cherries
Example: Simon and Lucas's Money
Simon has 5 times as much money as Lucas. They have a total of $q.
(a) Express the amount of money Lucas has in terms of q.
Total units = 5 + 1 = 6 units
6 units = $q
1 unit = $q ÷ 6 = $
(b) Find the amount of money Lucas has when q = 84.
Lucas's money = $
= $(84 ÷ 6)
= $14
Practice: Multiplication
If m = 8, find the value of 5m + 3.
ShowHide solution
5m + 3
= 5 × 8 + 3
= 40 + 3
= 43
4. Word Problem Evaluation
Example: Bus Rental
Mr Low rented a bus to take p students on a learning journey. He was charged $30 for rental of the bus and $6 for each student.
(a) Express the total amount Mr Low paid in terms of p.
Amount = $30 + $6 × p
= $30 + $6p
= $(30 + 6p)
(b) Find the total amount Mr Low paid when p = 20.
Amount = $(30 + 6p)
= $(30 + 6 × 20)
= $(30 + 120)
= $150
Practice: Word Problem
A taxi charges $3 flag-down fare plus $2 per kilometre. If the journey is k kilometres:
(a) Write the fare in terms of k.
(b) Find the fare when k = 12.
ShowHide solution
(a) Fare = $3 + $2k = $(3 + 2k)
(b) When k = 12:
Fare = $(3 + 2 × 12)
= $(3 + 24)
= $27
Practice: Division Expression
$w is shared equally among 4 friends. Each friend also receives an extra $5.
(a) Express how much each friend gets in terms of w.
(b) Find the amount each friend gets when w = 80.
ShowHide solution
(a) Each friend gets $w ÷ 4 + $5 = $
(b) When w = 80:
Each friend gets $(80 ÷ 4 + 5)
= $(20 + 5)
= $25
Steps for Evaluation
- Identify the expression (e.g., 4x + 9)
- Write down the given value (e.g., x = 15)
- Substitute - Replace the letter with the number
- Calculate - Follow order of operations (× ÷ before + -)
- Write the answer with units if needed
Key Takeaways
- ✓ Substitution = replacing a letter with a number
- ✓ When substituting, 4x means 4 × x
- ✓ means x ÷ 6
- ✓ Multiply and divide before you add and subtract (work out anything in brackets first)
- ✓ Check your answer makes sense in the context
- ✓ Include units in your final answer (cm, $, etc.)