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Simultaneous Equations

A self-marking, multi-level set of exercises on solving pairs of simultaneous equations.

Menu Level 1 Level 2 Level 3 Level 4 Level 5 Level 6 Level 7 Exam Help More

This is level 7: linear, quadratic and other pairs of simultaneous equations. You will be awarded a trophy if you get at least 9 correct and you do this activity online.

The following questions typically will have two pairs of possible answers. The system has chosen one of the pairs as the correct answer. First calculate both pairs and then try typing them in one at a time until you find the required pair (the Check button will let you know if you have the correct answer as chosen by the system).

\(x - y = 1\\y = x^2 -7\)

x= y= Correct Wrong

\(x - y = 6\\y = x^2 -78\)

x= y= Correct Wrong

\(b=a - 7\\b = a^2 -189\)

a= b= Correct Wrong

\(2c - d =12\\c^2 -75 = d\)

c= d= Correct Wrong


e= f= Correct Wrong


g= h= Correct Wrong


j= k= Correct Wrong

\(2m + 3n =9\\2m^2 + n + 3m=89\)

m= n= Correct Wrong

\(2p^2 - q + 5p=-4\\2p - 3q =-10\)

p= q= Correct Wrong

\(r^2+s^2=72\\r + s =12\)

r= s= Correct Wrong


u= v= Correct Wrong

\(w^2-z^2=-32\\3w - 2z =18\)

w= z= Correct Wrong


This is Simultaneous Equations level 7. You can also try:
Level 1 Level 2 Level 3 Level 4 Level 5 Level 6


Try your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help.

When you have got all of the questions correct you may want to print out this page and paste it into your exercise book. If you keep your work in an ePortfolio you could take a screen shot of your answers and paste that into your Maths file.

Why am I learning this?

Mathematicians are not the people who find Maths easy; they are the people who enjoy how mystifying, puzzling and hard it is. Are you a mathematician?

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Featured Activity

Practice Papers

Practice Papers

Mathematics GCSE(9-1) Higher style questions and worked solutions presented as twenty short, free, practice papers to print out.


There are answers to this exercise but they are available in this space to teachers, tutors and parents who have logged in to their Transum subscription on this computer.

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Go Maths

Learning and understanding Mathematics, at every level, requires learner engagement. Mathematics is not a spectator sport. Sometimes traditional teaching fails to actively involve students. One way to address the problem is through the use of interactive activities and this web site provides many of those. The Go Maths page is an alphabetical list of free activities designed for students in Secondary/High school.

Maths Map

Are you looking for something specific? An exercise to supplement the topic you are studying at school at the moment perhaps. Navigate using our Maths Map to find exercises, puzzles and Maths lesson starters grouped by topic.


If you found this activity useful don't forget to record it in your scheme of work or learning management system. The short URL, ready to be copied and pasted, is as follows:

Alternatively, if you use Google Classroom, all you have to do is click on the green icon below in order to add this activity to one of your classes.

It may be worth remembering that if should go offline for whatever reason, there is a mirror site at that contains most of the resources that are available here on

When planning to use technology in your lesson always have a plan B!

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Description of Levels



Projectable - A set of simultaneous equations designed to be shown one at a time to the whole class.

Level 1 - Equations that can be added or subtracted to eliminate one variable.

Level 2 - Equations that can be added or subtracted to eliminate one variable after one of the equations has been multiplied by a constant.

Level 3 - Equations that can be added or subtracted to eliminate one variable after both of the equations have been multiplied by constants.

Level 4 - Equations with two variables that are not written in the standard way.

Level 5 - Real life problems that can be solved by writing them as simultaneous equations.

Level 6 - Equations which include fractions in some way.

Level 7 - Linear, quadratic and other pairs of simultaneous equations.

These Level 7 questions will require you to be able to solve Quadratic Equations.

Exam Style Questions - A collection of problems in the style of GCSE or IB/A-level exam paper questions (worked solutions are available for Transum subscribers).

More Simultaneous Equations including lesson Starters, visual aids, investigations and self-marking exercises.


There is a printable worksheet to go with this activity. It is an exercise that appeared in an algebra book published in 1895. It starts with basic questions but soon gets tricky!



Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

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Curriculum Reference

See the National Curriculum page for links to related online activities and resources.

Help Video

The examples used in the video are available to teachers as projectable slides.

Level 7 Example

\(x-y=1 \qquad \mathbf{A}\\y=x^2-21 \qquad \mathbf{B}\)

One way of solving this pair of simultaneous equations is to use the substitution method.
Equation \(\mathbf{A}\) can be rearranged to make \(x\) the subject

\(x = 1 + y\)

This expression for \(x\) can be substituted into equation \(\mathbf{B}\):


This is a quadratic equation that can be solved by factorisation.

\( (y-4)(y+5) = 0\)

Either \(y=4\) in which case, by substituting into equation \(\mathbf{A}\), \(x=5\)

Or \(y=-5\) in which case, by substituting into equation \(\mathbf{A}\), \(x=-4\)

These Level 7 questions will require you to be able to solve Quadratic Equations.

Alternate method


Get both equations into the "y=" format

\(y=x-1 \\y=x^2-21\)

Set the equations equal to each other

\(x^2-21 = x-1\)

Rearrange into the standard quadratic equation format

\(x^2-x-20 = 0\)

Solve the quadratic equation

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

\(x=5 \qquad or \qquad x=-4\)

Use the linear equation to get matching y values

When \(x = 5, \qquad y = 4\)
When \(x = -4, \qquad y = -5\)

You can check your answers by substituting them both into equation \( \mathbf{B} \) to see if it balances.

This example is not intended to teach you everything you need to know about this type of simultaneous equations. It is here as a reminder and is no substitute for your teacher or tutor.

Simultaneous Equations Checklist

  1. Decide if the equations are in the correct form.
  2. Decide if we need to manipulate one or both equations.
  3. Decide if we need to add or subtract.
  4. Successfully add or subtract algebraic expressions, possibly involving negative numbers.
  5. Solve a linear equation, possibly involving negative numbers.
  6. Substitute the solution into an algebraic expression.
  7. Solve another linear equation.
  8. Solve another linear equation.
  9. Substitute two solutions into an algebraic expression to check the answer.
  10. Interpret the solution.

These steps are developed and discussed in "How I Wish I'd Taught Maths: Lessons learned from research, conversations with experts, and 12 years of mistakes" by Craig Barton

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly. You can double-click the 'Check' button to make it float at the bottom of your screen.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

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