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 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.

E.g. Linear simultaneous equations

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

E.g. Quadratic simultaneous equations

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

What is rearranging equations?

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

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

Equations worksheet

Get your free equations worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOON
x

Equations worksheet

Get your free equations worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOON

Solving equations

Types of 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


Equations - Linear equations 1a

E.g.


Solve: 4x+7=2x+12


Equations - Linear equations 1b

E.g.


Solve: 3(x-5)=18


Equations - linear equations 1c


OR


Equations - linear equations 1c (2) 1

E.g.


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


Equations - linear equations 1d


  • Step by step guide: Linear equations – with fractions (coming soon)

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


Equations - linear equations 1e

Quadratic equations

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

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}


\begin{aligned} x-2&=0 \quad \text{or} \quad x-3=0\\\\ x&=2 \quad \text{or} \quad x=3 \\\\ \end{aligned}


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 equations – quadratic formula (coming soon)

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: Quadratic equations – completing the square (coming soon)

E.g.


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


Equations - quadratics 2d


The solutions are to be found where y=0


x=2 \quad \textrm{or} \quad x=4


  • Step by step guide: Quadratic equations – graphically (coming soon)

Simultaneous equations

E.g.


Solve these simultaneous equations:


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


Equations - Simultaneous linear equations 3a


E.g.


Solve these simultaneous equations:


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


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


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.


Equations - Simultaneous linear equations 3c


The solution is x=1.5 and y=2


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}

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
GCSE Quiz True

x=10
GCSE Quiz False

x=6
GCSE Quiz False

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

2. Solve:

 

2(x+5)=3x+4

x=4
GCSE Quiz False

x=5
GCSE Quiz False

x=6
GCSE Quiz True

x=3
GCSE Quiz False
\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
GCSE Quiz False

x=25
GCSE Quiz True

x=27
GCSE Quiz False

x=29
GCSE Quiz False
\begin{aligned} \frac{x-4}{3}&=7\\\\ x-4&=21\\\\ x&=25 \end{aligned}

4. Solve:

 

x^2+6x+8=0

x=2 \quad \text{or} \quad x=4
GCSE Quiz False

x=8 \quad \text{or} \quad x=6
GCSE Quiz False

x=-8 \quad \text{or} \quad x=-6
GCSE Quiz False

x=-2 \quad \text{or} \quad x=-4
GCSE Quiz True
\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}

x=4 \quad \text{and} \quad y=2
GCSE Quiz False

x=4 \quad \text{and} \quad y=1
GCSE Quiz True

x=3 \quad \text{and} \quad y=1
GCSE Quiz False

x=3 \quad \text{and} \quad y=2
GCSE Quiz False
\begin{aligned} 2x+5y&=13 \quad [1]\\ x+2y&=6 \quad [2]\\ \\ [1]\times2 \quad 4x+10y&=26 \quad [3]\\ [2]\times5 \quad 5x+10y&=30 \quad[4]\\ \hline [4]-[3] \qquad \qquad x&=4\\ \\ \text{substitute} \quad x=4 \quad \text{into} \quad[2]\\ 4+2y&=6\\ 2y&=2\\ y&=1\\ \\ \text{solution is} \quad x=4 \quad \text{and} \quad y=1 \end{aligned}

6. Solve:

 

V=\pi r^2h

h=V-\pi r^2
GCSE Quiz False

h=\frac{\pi V}{r^2}
GCSE Quiz False

h=\frac{V}{\pi r^2}
GCSE Quiz True

h=\sqrt{\frac{V}{\pi r}}
GCSE Quiz False
\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)

Show answer
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)

Show answer

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)

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

(1)

 

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

(1)

 
x=5 \quad \text{or} \quad x=-4

(1)

4. Rearrange the equation to make x the subject:

 

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

 

(2 marks)

Show answer
\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

The next lessons are

  • Solving simple linear equations
  • Solving quadratics
  • Solving simultaneous equations
  • Rearranging equations
  • Inequalities
  • Finding approximate solutions to equations numerically using iteration

Still stuck?

Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.

Find out more about our GCSE maths revision programme.