# Substitution

GCSELevel 4-5AQAEdexcelOCRWJEC

## Substitution

Substitution means replacing letters or symbols in algebraic equations or expressions with numbers to find out their total value.

Level 4-5GCSEAQAEdexcelOCRWJEC

## Using Substitution

Substitution can be helpful in utilising formulae and equations.

For example, we know

$\text{Speed} = \dfrac{\text{Distance}}{\text{Time}}$

To utilise this helpful equation, we need to use substitution:

It takes Esme $45\text{ minutes}$ to travel to school, which is $3\text{ km}$ away. We can use substitution into the equation above the find the speed Esme travels in $\text{km/h}$:

Distance $= 3\text{ km}$

Time $= 0.75\text{ hours}$

$\text{Speed} = \dfrac{3}{0.75}$

$\text{Speed} = 4\text{ km/h}$

Level 4-5GCSEAQAEdexcelOCRWJEC

@mmerevise

Level 4-5GCSEAQAEdexcelOCRWJEC

## Example 1

Given

$k = 5a + 19b - 12c - \dfrac{d}{4}$

Work out the value of $k$ when

$a=8\\$ $b=2\\$ $c=4\\$ $d=48$

[2 marks]

Substituting the values of $a, b, c,$ and $d$ into the equation:

$k = 5\times8 + 19\times2 - 12\times4 - \dfrac{48}{4}$

You must remember BIDMAS when using substitution, so division/multiplication must come before addition/subtraction.

$k = 40 + 38 - 48 - 12$

$k =18$

Level 4-5GCSEAQAEdexcelOCRWJEC

## Example 2

Using the quadratic formula is an example of substitution

$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$

Solves equations in the form

$ax^2+bx+c=0$

Use substitution into the quadratic formula to solve

$3x^2+5x+2=0$

[3 marks]

We can see from the equation that:

$a=3\\$ $b=5\\$ $c=2$

So, we can substitute these into the quadratic equation:

$x=\dfrac{-5\pm\sqrt{5^2-4(3)(2)}}{2(3)}$

Which gives us the solutions $x=-\dfrac{2}{3}$ and $x=-1$

Level 4-5GCSEAQAEdexcelOCRWJEC

## Substitution Example Questions

If $a=36$, $b=\dfrac{1}{2}(36) = 18$

Substitution:

$Z=2(36)+5(18)^3\\$ $Z=29232$

Gold Standard Education

Substitution:

$46 = 5m + 16$

Rearrange and solve:

$46-16=5m\\$ $30=5m\\$ $m=6$

Gold Standard Education

Substitution:

$y = 2(78)-21\\$ $y = 135$

Ezra would be expected to score $135$ in his English test.