We have seen we can form expressions using numbers, mathematical operations and variables. If an expression contains a variable and we replace the variable with a particular value or expression, this is called substitution. For example, if we had $4$4 full boxes of matches and $12$12 additional loose matches, then the expression $4m+12$4m+12 would give us the total number of matches where $m$m was the number of matches in a full box. If we were then told the additional information that there are $50$50 matches in a full box, we could evaluate the expression to find the total number of matches by making the substitution $m=50$m=50 in the expression:
| $4m+12$4m+12 | $=$= | $4\times50+12$4×50+12 |
| $=$= | $200+12$200+12 | |
| $=$= | $212$212 |
Worked example
Example 1
If $x=3$x=3, evaluate the expression $6x-4$6x−4.
Think: This means that everywhere the letter $x$x has been written, we will replace it with the number $3$3.
Do:
| $6x-4$6x−4 | $=$= | $6\times3-4$6×3−4 |
| $=$= | $18-4$18−4 | |
| $=$= | $14$14 |
Example 2
If $x=6$x=6 and $y=0.5$y=0.5, evaluate the expression $6x-2y-12$6x−2y−12.
Think: The same process applies even if there is more than one unknown value, we will replace the letter $x$x with the number $6$6, and the letter $y$y with the number $0.5$0.5. We also need to keep the order of operations in mind when we do these kinds of calculations!
Do:
| $6x-2y-12$6x−2y−12 | $=$= | $6\times6-2\times0.5-12$6×6−2×0.5−12 |
Replacing $x$x with $6$6, and $y$y with $0.5$0.5. |
| $=$= | $36-1-12$36−1−12 |
Evaluating multiplication before subtraction. |
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| $=$= | $23$23 |
Example 3
If $a=3$a=3 and $b=-4$b=−4, evaluate the expression $a\left(10-2b\right)$a(10−2b).
Think: Just like before, we will replace the letter $a$a with the number $3$3, and the letter $b$b with the number $-4$−4. To avoid confusion with the operations in the expression we will place the negative number within brackets.
Do:
| $a\left(10-2b\right)$a(10−2b) | $=$= | $3\left(10-2\times\left(-4\right)\right)$3(10−2×(−4)) |
Replace $a$a with $3$3, and $b$b with $\left(-4\right)$(−4). |
| $=$= | $3\left(10+8\right)$3(10+8) |
Simplify the terms within the bracket. |
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| $=$= | $3\left(18\right)$3(18) |
Evaluate the bracket before multiplication. |
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| $=$= | $54$54 |
When making a substitution and evaluating an expression be careful to follow order of operations, just as we did in our first chapter.
When substituting a negative value, place brackets around the value so the sign is not confused with operations in the expression.
Use the following applet to practise evaluating an expression given the value of $x$x.
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Practice questions
question 1
Evaluate $8x+4$8x+4 when $x=2$x=2.
Question 2
If $m=-3$m=−3 and $n=4$n=4, evaluate the following:
$mn-\left(m-n\right)$mn−(m−n)
$m^2+9n$m2+9n
Question 3
Evaluate $\frac{2a\times9}{5b}$2a×95b when $a=25$a=25 and $b=-2$b=−2.
Find the exact value in simplest form.