Substitution (Cambridge International IGCSE Maths)

Substitution

What is substitution?

• Substitution means replacing a letter (variable) in a formula with a given number

How do I substitute numbers into a formula?

• Write down the formula

• Replace (substitute) the letters in the formula with the given numbers

  • If substituting in a negative number, it is important to put brackets around it

    • For example, (-3)

• Simplify any numerical calculations

• Calculate the final value

• Sometimes the result is an equation which you can then solve

Exam Hint On your calculator, don't forget to put brackets around any substituted negative numbers!

Be careful with negatives and powers. For example, if $x=-4$ then $x^{2} = ( - 4 )^{2} = 16$.

Worked Example (a) Given that $x=2$ and $y=-4$, find the value of the expression:

(i) $2 x ( x + 3 y )$

(ii) $y^{2}+y^{3}$

Answer:

(i)

Substitute the numbers given Use brackets () around negative numbers

$\begin{gathered} 2 \times 2 \times ( 2 + 3 \times ( - 4 ) ) \end{gathered}$

Complete the calculation Show every step of working, following the order of operations correctly

$\begin{gathered} = 2 \times 2 \times ( 2 - 12 ) \\ = 2 \times 2 \times ( - 10 ) \\ = 4 \times ( - 10 ) \end{gathered}$

-40

(ii)

Substitute the numbers given Use brackets () around negative numbers

$( - 4 )^{2} + ( - 4 )^{3}$

Work out the powers remembering to include the brackets

$16 + ( - 64 )$

Complete the calculation

$16-64$

Final answer: -48

(b) The formula $P=2l+2w$ is used to find the perimeter, $P$, of a rectangle of length $l$ and width $w$.

Given that the rectangle has a perimeter of 20 cm and a width of 4 cm, find its length.

Answer:

Substitute the values you are given for $P$ and $w$ into the formula

$20=2\times l+2\times4$

Simplify

$20=2l+8$

Solve the resulting equation to find the value of $l$ Start by subtracting 8 from both sides

$12=2l$

Divide both sides by 2

Final answer: $l=6\mathrm{cm}$