Simultaneous Equations Questions, Revision and Worksheets

Simultaneous Equations Questions, Revision and Worksheets

GCSE 4 - 5GCSE 6 - 7AQAEdexcelOCRWJECAQA 2022Edexcel 2022OCR 2022WJEC 2022

Simultaneous Equations 

Simultaneous equations are multiple equations that share the same variables and which are all true at the same time.

When an equation has 2 variables its much harder to solve, however, if you have 2 equations both with 2 variables, like

2x+y=10\,\,\,\text{ and }\,\,\,x+y=4

then there is a solution for us to find that works for both equations. These equations are called simultaneous for this reason.

There are 2 main types of equation you need to be able to solve.

Make sure you are happy with the following topics before continuing.

Level 4-5 GCSE AQA Edexcel OCR WJEC

Type 1: Linear Simultaneous equations.

To do this, we’ll use a process called elimination – we’re going to eliminate one of the variables by subtracting one equation from the other. We will write one equation on top of the other and draw a line underneath, as with normal subtraction.

Example: Find the solution to the following simultaneous equations.

4x + 3y = 14 \,\,\,\,,\ 5x+7y=11

Step 1: Write one equation above the other.

Both equations need to be in the form ax+by=c, so rearrange if needed.

\begin{aligned}4x + 3y &= 14 \\ 5x+7y &= 11 \end{aligned}

Step 2: Get the coefficients to match

The coefficients are the numbers before x and y, make the x coefficients the same by scaling up both equations

(\times5) \,\,\,\,\,\,\,\,\,4x + 3y = 14\,\,\, gives \,\,\, 20x + 15y = 70

\,\, (\times4)\,\,\,\,\,\,\,\,\, 5x+7y = 11 \,\,\, gives \,\,\, 20x+28y=44

Step 3: Add or subtract the equations to eliminate terms with equal coefficients.

As both equations are +20x we must subtract the equations.

\begin{aligned}20x + 15y &= 70 \\ (-)\,\,\,\,\,\,\,\,\, 20x+28y&=44 \\ \hline-13y&=26\end{aligned}

Step 4: Solve the resulting equation

\begin{aligned}(\div-13)\,\,\,\,\,\,\,\,\,-13y&=26 \\ y &=-2\end{aligned}

Step 5: Substitute the answer into the simplest of the two equations to find the other variable.

\begin{aligned}4x + 3y &= 14 \\ 4x +3(-2) &=14 \\ 4x -6 &= 14 \\ 4x &= 20 \\ x &= 5\end{aligned}

This gives the final answer to be;

x = 5, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, y = -2

Level 4-5 GCSE AQA Edexcel OCR WJEC
Level 6-7 GCSE AQA Edexcel OCR WJEC

Type 2: Non-linear Simultaneous Equations

Because one of these equations is quadratic (Non-linear), we can’t use elimination like before. Instead, we have to use substitution.

Example: Solve the following simultaneous equations.

x^2+2y=9,\,\,\,\,y-x=3

Step 1: Rearrange the linear equation to get one of the unknowns on its own and on one side of the equals sign.

\begin{aligned}(+x)\,\,\,\,\,\,\,\,\,y-x&=3 \\ y &= x+3\end{aligned}

Step 2: Substitute the linear equation into the non-linear.

 We know y=x+3 so we can replace the y in the first equation with x+3:

\begin{aligned} x^2+2y&=9 \\ x^2+2(x+3)&=9\end{aligned}

Step 3: Expand and solve the new quadratic formed.

\begin{aligned}x^2+2(x+3)&=9 \\ x^2 + 2x+6 &= 9 \\ x^2+2x-3&=0 \\ (x-1)(x+3)&=0\end{aligned}

x=1 \,\,\, and \,\,\,x=-3.

Step 4:  Substitute both values back into the simplest equation to find both versions of the other variable.

\text{When }x=1, \,\,\,\,\,\,\,\,\, y=1+3=4,

\text{When }x=-3, \,\,\,\,\,\,\,\,\, y=-3+3=0.

Thus, we have two solution pairs:

x=1, y=4, and x=-3, y=0.

Level 8-9 GCSE AQA Edexcel OCR WJEC
Level 4-5 GCSE AQA Edexcel OCR WJEC

Example: Applied Simultaneous Equation Question

A store sells milkshakes and ice creams.

2 milkshakes and 2 ice creams, costs £7

4 milkshakes and 3 ice creams, costs £12

Work out the cost of an individual milkshake and an individual ice cream.

[4 marks]

Step 1: we need to do is form the two simultaneous equations.

Let’s say that the price of a milkshake is a, and the price of an ice cream is b.

This creates the following two equations.

2a+2b=7

4a+3b=12

Step 2: Now we must get the coefficients to match, in this case we can multiply the first equation by 2

(\times 2) \,\,\,\,\,\,\,\,\, 2a+2b=7 \,\,\, \text{Gives} \,\,\, 4a + 4b = 14

Step 3: Subtract the equations to eliminate terms with equal coefficient.

\begin{aligned}4a + 4b &= 14 \\ (-)\,\,\,\,\,\,\,\,\, 4a+3b&=12 \\ \hline b&=2\end{aligned}

Step 4: Substitute the answer into the simplest of the two equations to find the other variable.

\begin{aligned}2a+2b&=7 \\ 2a + 2(2) &=7 \\ 2a+4&=7 \\ 2a &=3 \\ a& = 1.5\end{aligned}

This gives the final answer to be

Milkshake (a) = £1.50

Ice Cream (b) = £2.00

Level 4-5 GCSE AQA Edexcel OCR WJEC

Example Questions

Straight away we can subtract equation 2 from equation 1 so that,

 

\begin{aligned}y&=2x-6\\ y&=\dfrac{1}{2}x+6 \\ \\ (y-y)&=(2x-\dfrac{1}{2}x)-6-6 \\ 0&= \dfrac{3}{2}x-12 \end{aligned}

 

If we rearrange to make x the subject we find,

 

x=\dfrac{2\times12}{3}=\dfrac{24}{3}=8

 

Substituting x=8 back into the original first equation,

 

\begin{aligned}y&=2(8)-6 \\ y&=10\end{aligned}

Hence, the solution is,

x=8, y=10

If we multiply the second equation by 2, we have two equations both with a 2x term, hence subtracting our new equation 2 from equation 1 we get,

 

\begin{aligned}2x-3y&=16\\ 2x+4y&=-12 \\ \\ (2x-2x)+(-3y-4y)&= 16-(-12) \\ 0x -7y &=28\end{aligned}

 

If we rearrange to make y the subject we find,

 

y=\dfrac{28}{-7}=-4

 

Substituting y=-4 back into the original second equation,

 

\begin{aligned}x+2(-4)&=-6 \\ x-8&=-6 \\ x&=2\end{aligned}

Hence, the solution is,

x=2, y=-4

If we multiply the first equation by 3, we have two equations both with a 3x term, hence subtracting our new equation 2 from equation 1 we get,

 

\begin{aligned}3x+6y+30&=0 \\ 3x-5y-14&=0 \end{aligned}

(3x-3x)+(6y-(-5y))+(30-(-14)) = 0

\begin{aligned} 11y &= -44 \\ y &= - 4\end{aligned}

 

Substituting y=-4 back into the original first equation,

 

\begin{aligned}x+2(-4)+10&=0\\ x-8+10 &=0 \\ x&=-2 \end{aligned}

Hence, the solution is,

x=-2, y=-4

Let A be the cost of an adult ticket and let C be the cost of a child ticket, thus we have two simultaneous equations,

 

\begin{aligned} 2A+3C&=20 \\ A+C&=8.5 \end{aligned}

 

If we multiply the second equation by 2, we have two equations both with a 2A term, hence subtracting our new equation 2 from equation 1 we get,

 

\begin{aligned} 2A+3C&=20 \\ 2A+2C&=17 \\ \\ (2A-2A)+(3C-2C) &=(20-17) \\ C&=3\end{aligned}

 

Then, substituting this value back into the original equation 2, we get,

 

\begin{aligned} A+3&=8.5 \\ A & =5.5\end{aligned}

 

Therefore, the cost of a child ticket is £3, and the cost of an adult ticket is £5.50.

If we multiply the first equation by 2, we have two equations both with a 2y term, hence adding our new equation 1 and equation 2 we get,

 

\begin{aligned} 2x^2-2y&=28 \\ 2y-4&=12x \\ \\ 2x^2+(-4)+(-2y+2y)&=28+12x \\ 2x^2-12x-32&=0\end{aligned}

After rearranging to form a quadratic we can solve for x,

 

\begin{aligned} 2x^2-12x-32&=0 \\ 2(x^2-6x-16)&=0 \\ (x-8)(x+2)&=0\end{aligned}

Therefore, the 2 solutions for x are x=8, x=-2. To find the two y solutions, we can substitute these values back into the original second equation.

 

When x=8,

\begin{aligned}2y-4&=96 \\ y&=50 \end{aligned}

When x=-2,

\begin{aligned}2y-4&=-24 \\ y &= -10\end{aligned}

Thus, the two pairs of solutions are,

x=8,y=50 and x=-2,y=-10

Related Topics

MME

Solving Equations

Level 4-5GCSEKS3
MME

Rearranging Formulae

Level 4-5Level 6-7GCSEKS3
MME

Solving Quadratic Equations Through Factorising

Level 4-5GCSE
MME

The Quadratic Formula

Level 6-7GCSE

Worksheet and Example Questions

Site Logo

(NEW) Simultaneous Equations (Linear) Exam Style Questions - MME

Level 4-5 GCSENewOfficial MME
Site Logo

(NEW) Simultaneous Equations (Non-linear) Exam Style Questions - MME

Level 8-9 GCSENewOfficial MME

Drill Questions

Site Logo

Solving Simultaneous - Drill Questions

Level 4-5 GCSE
Site Logo

Simultaneous Equations (Linear) - Drill Questions

Level 4-5 GCSE
Site Logo

Simultaneous Equations (Linear and Non-linear) - Drill Questions

Level 8-9 GCSE
Site Logo

Simultaneous Equations (Linear and Non-linear) 2 - Drill Questions

Level 8-9 GCSE
Site Logo

Simultaneous Equations (Mixed)

Level 8-9 GCSE

You May Also Like...

GCSE Maths Revision Cards

Revise for your GCSE maths exam using the most comprehensive maths revision cards available. These GCSE Maths revision cards are relevant for all major exam boards including AQA, OCR, Edexcel and WJEC.

From: £8.99
View Product

GCSE Maths Revision Guide

The MME GCSE maths revision guide covers the entire GCSE maths course with easy to understand examples, explanations and plenty of exam style questions. We also provide a separate answer book to make checking your answers easier!

From: £14.99
View Product

GCSE Maths Predicted Papers 2022 (Advance Information)

GCSE Maths 2022 Predicted Papers are perfect for preparing for your 2022 Maths exams. These papers have been designed based on the new topic lists (Advance Information) released by exam boards in February 2022! They are only available on MME!

From: £5.99
View Product

Level 9 GCSE Maths Papers 2022 (Advance Information)

Level 9 GCSE Maths Papers 2022 are designed for students who want to achieve the top grades in their GCSE Maths exam. Using the information released in February 2022, the questions have been specifically tailored to include the type of level 9 questions that will appear in this year's exams.

£9.99
View Product