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6.02 Substitution into expressions


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$6x4.

Think: This means that everywhere the letter $x$x has been written, we will replace it with the number $3$3.


$6x-4$6x4 $=$= $6\times3-4$6×34
  $=$= $18-4$184
  $=$= $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!


$6x-2y-12$6x2y12 $=$= $6\times6-2\times0.5-12$6×62×0.512

Replacing $x$x with $6$6, and $y$y with $0.5$0.5.

  $=$= $36-1-12$36112

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.


$a\left(10-2b\right)$a(102b) $=$= $3\left(10-2\times\left(-4\right)\right)$3(102×(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  



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.



substitute numerical values into algebraic expressions; for example, substitute different values of x to evaluate the expressions 3x/5, 5(2x-4)

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