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VCE 11 Methods 2023

1.03 Substitution

Lesson

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$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$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!

Do:  

$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.

Do:  

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

 

Remember!

When making a substitution and evaluating an expression be careful to follow order of operations.

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

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 3

Find the value of the expression when $x=-3$x=3:

$-3x^3+4x^2+7x+5$3x3+4x2+7x+5

 

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=13b2h, 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$13b2h

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

The perimeter of a square with side lengths of $a$a is given by the formula $P=4\times a$P=4×a.

Find $P$P if the length of each side is $5$5 cm.

question 5

The area of a triangle is given by the formula $A=\frac{1}{2}$A=12$($(base$\times$×height$)$).

If the base of a triangle is $3$3 cm and its height is $10$10 cm, find its area.

Question 6

The simple interest generated by an investment is given by the formula $I=\frac{P\times R\times T}{100}$I=P×R×T100.

Given that $P=1000$P=1000, $R=6$R=6 and $T=7$T=7, find the interest generated.

Question 7

Energy can be measured in many forms. A quantity of energy is given in units of Joules (J).

The kinetic energy, $E$E, of an object in motion is calculated using the following formula:

$E=\frac{mv^2}{2}$E=mv22

where $m$m is the mass of the object in kilograms and $v$v is the speed of the object in metres per second.

  1. Find the kinetic energy, $E$E, of an object with a mass of $6$6 kg, travelling at a speed of $19$19 metres per second.

 

Outcomes

U1.AoS2.12

substitute integer, simple rational and irrational numbers in exact form into expressions, including rules of functions and relations, and evaluate these by hand

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