 6.02 Substitution into expressions

Lesson

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 number, 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 examples

Example 1

If $x=3$x=3, evaluate the expression $6x-4$6x4.

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$6x2y12.

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. $=$= $23$23
Example 3

If $a=3$a=3 and $b=-4$b=4, evaluate the expression $a\left(10-2b\right)$a(102b).

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. $=$= $3\left(18\right)$3(18) Evaluate the bracket before multiplication. $=$= $54$54

Remember!

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.

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:

1. $mn-\left(m-n\right)$mn(mn)

2. $m^2+9n$m2+9n

Question 3

Evaluate $\frac{2a\times9}{5b}$2a×95b when $a=25$a=25 and $b=-2$b=2.

1. Find the exact value in simplest form.

Question 4

Evaluate $7a^2$7a2 for $a=-3$a=3.

Outcomes

1.2.2

substitute values for the variables in a mathematical formula in given form to calculate the value of the subject of the formula