# Substitution

## Substitution Revision

**Substitution**

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

**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}

**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

**Example 2**

Using the **quadratic formula** is an example of **substitution**.

Given the quadratic formula

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=2So, 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

## Substitution Example Questions

**Question 1: **

Z=2a+5b^3

Find the value of Z when a=36 and b=\dfrac{1}{2}a.

**[2 marks]**

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

Substitution:

Z=2(36)+5(18)^3\\ Z=29232**Question 2:**

P = 5m + 16

P = 46

Work out the value of m

**[2 marks]**

Substitution:

46 = 5m + 16Rearrange and solve:

46-16=5m\\ 30=5m\\ m=6**Question 3: **A teacher creates an equation to estimate the English exam scores of her students based on their Science scores.

y=2x-21

Where x is their Science score, and y is the estimate for the English score.

Ezra got 78 in his Science exam. Predict his English result.

**[2 marks]**

Substitution:

y = 2(78)-21\\ y = 135Ezra would be expected to score 135 in his English test.