4.02 Substitution

Substitution into expressions

We have seen we can form expressions using numbers, mathematical operations and variables. If an expression contains a variable we can replace the variable with a particular value or expression and 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.
	$=$=	$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))	Replacing $a$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

 

 

Practice questions

question 1  

Question 2

question 3

 

Substitution in formulas

Using algebra we can write relationships between different quantities concisely in formulas.

For example, instead of writing, "The area of a rectangle is calculated by multiplying the length by the width," we can write the algebraic formula $A=LW$A=LW. Formulae give a general rule relating variables and we can use substitution to evaluate the formula for specific cases, just as we substituted particular values for length and width when finding areas of rectangles. 

There are lots of topics that have common formulae: simple interest, perimeter, area, volume, temperature conversions and many more across a broad range of areas in Mathematics, Physics, Chemistry, finance and so forth. It is a good idea to be familiar with the formulae available for use in any topic before you start doing exercises.

Worked example

The volume of a square based pyramid is given by the formula $V=\frac{1}{3}b^2h$V=13​b2h, where $b$b is the length of a side of the base square and $h$h is the perpendicular height. 

Find the volume of a square based pyramid with $b=6$b=6 m and $h=5$h=5 m.

Think: Write out the formula, substitute in the known values and evaluate. 

Do:

$V$V	$=$=	$\frac{1}{3}b^2h$13​b2h	Write out the formula.
	$=$=	$\frac{1}{3}\times6^2\times5$13​×62×5 m3	Replace $b$b with $6$6 and $h$h with $5$5.
	$=$=	$\frac{1}{3}\times36\times5$13​×36×5 m3	Evaluate the term with the power and then multiply.
	$=$=	$60$60 m3	Evaluate and don't forget units for questions in context.

 

Practice questions

question 4

question 5

Question 6

Question 7