GCSE Maths Algebra

Equations

# Equations

Here we will learn about equations, including solving equations, linear equations, quadratic equations, simultaneous equations and rearranging equations.

There are also equations worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

## What are equations?

Equations are mathematical expressions which contain a variable and an equals sign.

We can solve an equation to find the value of the variable.

E.g.

\begin{aligned} &3x-5=7 \\\\\\ &\frac{4(x-2)}{5}=8 \\\\\\ &x^2=9 \\\\\\ &2x^2-3x-5=0 \end{aligned}

### What are equations? ### What are simultaneous equations?

Simultaneous equations are a pair of equations with two variables.  They can be solved to find a pair of values which make both equations true at the same time.

Step-by-step guide: Simultaneous equations

E.g. Linear simultaneous equations

\begin{aligned} x+y&=10\\\\ x-y&=4 \end{aligned}

E.g. Quadratic simultaneous equations

y=x^2-6x+8

y=2x+1

### What is rearranging equations?

Rearranging equations means we change the subject of the equation to display it in a different way.

Step-by-step guide: Rearranging equations

E.g.

The subject of the following equation is currently y .

y=3x+2

We can rearrange the equation to make x the subject:

x=\frac{y-2}{3}

Sometimes an equation is a formula. This is when it is used to solve a specific problem.

E.g.

Here is the formula to work out the area of a circle:

A=\pi r^2

It could be rearranged to find the radius if we are given the area:

r=\sqrt{\frac{A}{\pi }}

### Equations and identities

Equations and identities look very similar, but are two different things.

An identity is true for all values of the variable, whereas the equation is true for only certain values.

For example, the equation:

3x+5=14

can be solved, so it is true only when x=3.

However the identity:

3(x+2)\equiv3x+6

It can not be solved. The left-hand side always equals the right-hand side for all values of x.

### Identity sign

For identities there is a special symbol that can be used ‘≡’. It is like an equals sign =, but it means ‘identical to’.

### Iteration

Iteration is a process in which the approximate solution is determined using an iterative formula and a known starting number. When using iteration to calculate a root of an equation, the values converge towards a fixed point.

## Solving equations

### Types of equations ### Linear equations

Simple linear equations are solved by using a “balancing method” – doing the same operations to both sides of the equation. They can be checked by substituting the answer back into the original equation.

Step-by-step guide: Linear equations

E.g.

Solve: 3x+4=16

E.g.

Solve: 4x+7=2x+12

E.g.

Solve: 3(x-5)=18

OR

E.g.

Solve: \frac{5}{x}=10

Whilst these are NOT linear equations they are included here as the method used to solve them is very similar.

E.g.

Solve: 4x^2+1=35

Step-by-step guide: Equations with fractions

Quadratic equations are solved by using different methods. They can be checked by substituting the answer back into the original equation.

Step-by-step guide Quadratic equations

We factorise the quadratic and then solve each factor being equal to 0 , one at a time.

E.g.

Solve: x^2-5x+6=0

\begin{aligned} x^2 -5x+6 &= 0\\\\ (x-2)(x-3) &= 0 \\\\ \end{aligned}

Step-by-step guide: Solving quadratic equations by factorising

We can use the quadratic formula to solve any quadratic equation of the form

ax^2+bx+c

The quadratic formula is: x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}

E.g.

Solve: 3x^2-4x-5=0

Give your answers to 2 decimal places

\begin{aligned} &3x^{2}-4x-5=0\\\\ x&=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\ a&=3, \quad b=-4, \quad c=-5\\\\ x&=\frac{-(-4)\pm\sqrt{(-4)^2-4(3)(-5)}}{2(3)}\\\\ x&=\frac{4\pm\sqrt{76}}{6} \\\\ x&=2.1196... \quad \text{or} \quad x=-0.78629...\\\\ x&=2.12 \quad \text{or} \quad x=-0.79 \end{aligned}

Step-by-step guide: Quadratic formula

E.g.

Solve: x^2-4x-5=0

\begin{aligned} x^2 -4x-5 &= 0\\\\ x^2-4x&=5\\\\ (x-2)^2 -4&=5\\\\ (x-2)^2 &= 9\\\\ x-2 &= \pm 3\\\\ x&=2\pm 3\\\\ x=5 \quad &\textrm{or} \quad x=-1 \end{aligned}

Step-by-step guide: Completing the square

E.g.

Using the graph of y=x^2-6x+8 , solve: x^2-6x+8=0

The solutions are to be found where y=0

Step-by-step guide: Solving quadratic equations graphically

### Simultaneous equations

E.g.

Solve these simultaneous equations:

\begin{aligned} 2x+3y&=14\\\\ 6x-y&=12 \end{aligned}

Step-by-step guide: Simultaneous equations

E.g.

Solve these simultaneous equations:

\begin{aligned} y&=x^2-6x+8\\\\ y&=2x+1 \end{aligned}

\begin{aligned} x^2-6x+8&=2x+1\\ x^2-8x+7&=0\\ (x-7)(x-1)&=0\\ \\ x=7 \quad &\text{or} \quad x=1\\ \\ \text{substitue} \quad x=7 \quad \text{into} \quad y&=2x+1\\ y&=2(7)+1=15\\ \\ \text{substitue} \quad x=1 \quad \text{into} \quad y&=2x+1\\ y&=2(1)+1=3\\ \\ \text{solutions are} \quad x=7,\quad y=15& \quad \text{or} \quad x=1, \quad y=3\\ \end{aligned}

Step-by-step guide: Quadratic simultaneous equations

E.g.

Draw graphs to solve the simultaneous equations:

\begin{aligned} 2x+y=5 \\\\ y=2x-1 \end{aligned}

Draw the graphs on the same axes and see where they intersect.

The solution is x=1.5 and y=2

Step-by-step guide: Solving simultaneous equations graphically

### Rearranging equations

Example:

Make t the subject of: v=u+at

\begin{aligned} &v=u+at \\\\ &v-u=at\\\\ &\frac{v-u}{a}=t\\\\ &\text{Therefore} \\\\ &t= \frac{v-u}{a} \end{aligned}

Step-by-step guide: Make x the subject

### Common misconceptions

• Solutions to equations are not always positive whole numbers

Solutions to equations can be positive integers (whole numbers), but they can also be negative.  They can also be decimals or fractions.

• Don’t try to just write down the answer

When working with equations, take each step one at a time.  Keep your workings neat and tidy so you are more likely to be correct.

• If you multiply both sides (or divide) make sure that you do this to each and every term

When you are working with an equation and you need to multiply (or divide), it is easy to make a mistake.  Make sure you apply the multiplication to every term.

For example:

Each term on both sides is multiplied by 3

\begin{aligned} \frac{x}{3}&=4x-5\\\\ x&=12x-15 \end{aligned}

### Practice equations questions

1. Solve:

4x+10=30

x=5 x=10 x=6 x=9 \begin{aligned} 4x+10&=30\\\\ 4x&=20\\\\ x&=5 \end{aligned}

2. Solve:

2(x+5)=3x+4

x=4 x=5 x=6 x=3 \begin{aligned} 2(x+5)&=3x+4\\\\ 2x+10&=3x+4\\\\ 10&=x+4\\\\ 6&=x\\\\ \text{therefore} \quad x&=6 \end{aligned}

3. Solve:

\frac{x-4}{3}=7

x=31 x=25 x=27 x=29 \begin{aligned} \frac{x-4}{3}&=7\\\\ x-4&=21\\\\ x&=25 \end{aligned}

4. Solve:

x^2+6x+8=0    \begin{aligned} x^2+6x+8&=0\\\\ (x+2)(x+4)&=0\\\\ x=-2 \quad \text{or} \quad x&=-4 \end{aligned}

5. Solve:

\begin{aligned} 2x+5y=13 \\\\ x+2y=6 \end{aligned}    6. Rearrange correctly to make h the subject:

V=\pi r^2h

h=V-\pi r^2 h=\frac{\pi V}{r^2} h=\frac{V}{\pi r^2} h=\sqrt{\frac{V}{\pi r}} \begin{aligned} V &= \pi r^2 h\\\\ \frac{V}{\pi} &= r^2h\\\\ \frac{V}{\pi r^2} &=h\\\\ \text{therefore} \quad h&= \frac{V}{\pi r^2} \end{aligned}

### Equations GCSE questions

1. Solve:

3x-4=12

(2 marks)

3x=16

(1)

x=5\frac{1}{3}

(1)

2. Solve the simultaneous equation.

\begin{aligned} 2x+y&= 15\\\\ x-y &= 6 \end{aligned}

(3 marks)

Add the 2 equations

2x+x=15+6

(1)

\begin{aligned} 3x&= 21\\\\ x &= 7 \end{aligned}

(1)

\begin{aligned} 2x+y&= 15\\\\ 2(7)+y &= 15\\\\ 14+y&=15\\\\ y&=1 \end{aligned}

(1)

3. Solve:

x^2-x-20=0

(3 marks)

\begin{aligned} x^2-x-20&=0\\\\ (x\pm a)(x\pm b) &= 0 \end{aligned}

(1)

(x-5)(x+4)=0

(1)

(1)

4. Rearrange the equation to make x the subject:

y=\frac{x}{4}-5

(2 marks)

\begin{aligned} y&=\frac{x}{4}-5\\\\ y+5&=\frac{x}{4} \end{aligned}

(1)

\begin{aligned} 4(y+5)&=x\\\\ x&=4(y+5) \end{aligned}

(1)

## Learning checklist

You have now learned how to:

• Use algebraic methods to solve linear equations in 1 variable
• Rearrange formulae to change the subject
• To find approximate solutions of simultaneous linear equations
• Solve quadratic equations {including those that require rearrangement} algebraically by factorising
• Solve quadratic equations by completing the square
• Solve quadratic equations by using the quadratic formula
• Solve quadratic equations – find approximate solutions using a graph
• Solve 2 simultaneous linear equations in 2 variables algebraically
• Solve 2 simultaneous equations in 2 variables – one linear and one quadratic algebraically
• Solve 2 simultaneous equations in 2 variables – find approximate solutions using a graph

## Still stuck?

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#### GCSE Maths Papers - November 2022 Topics

Practice paper packs based on the November advanced information for Edexcel 2022 Foundation and Higher exams.

Designed to help your GCSE students revise some of the topics that are likely to come up in November exams.