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Three Unknowns

Solve these sets of three simultaneous, linear equations to find the values of the variables

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This is level 1: Standard set of questions with the equations set out in a familiar way. You will be awarded a trophy if you get at least 15 answers correct and you do this activity online.


\(2x+y-3z= -2 \\ 3x+y+z= 17 \\ x-y+2z = 7\)

\(x = \) \(y = \) \(z = \)


\( 5a+2b+c=42 \\ 3a+4b+2c= 42 \\ a+5b+c=30\)

\(a = \) \(b = \) \(c = \)


\(2d+3e-4f = -18 \\ d-e-f= -7\\ 9d+2e-2f=8\)

\(d = \) \(e = \) \(f = \)


\( g-7h-7i=-53 \\ 2g-2h+i= -1\\ 5g+3h+i = 33\)

\(g = \) \(h = \) \(i = \)


\( j+k+l= 18 \\ 2j-3k+9l= 51\\ -j+k-3l=-20\)

\(j = \) \(k = \) \(l = \)


\( -m+4n-p=-12 \\ 2m-3n+5p= 56\\ 7m-n+p=57\)

\(m = \) \(n = \) \(p = \)


\( 2q+2r-5s=-11 \\ q-r+2s= 8\\ -q+5r+5s=-51\)

\(q = \) \(r = \) \(s = \)


\( t-5u-5v=-4 \\ 2t+u+3v= 17\\ -5t+5u-v=-42\)

\(t = \) \(u = \) \(v = \)


\( 9w+x-y=36 \\ 2w+7x-3y= -41\\ -7w+5x+5y=-50\)

\(w = \) \(x = \) \(y = \)


\( 8x-5y+2z=-19 \\ -3x+7y-z= -6\\ 2z-5x-y=-5\)

\(x = \) \(y = \) \(z = \)


This is Three Unknowns level 1. You can also try:
Level 2 Level 3


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?

Comment recorded on the 1 August 'Starter of the Day' page by Peter Wright, St Joseph's College:

"Love using the Starter of the Day activities to get the students into Maths mode at the beginning of a lesson. Lots of interesting discussions and questions have arisen out of the activities.
Thanks for such a great resource!"

Comment recorded on the 17 November 'Starter of the Day' page by Amy Thay, Coventry:

"Thank you so much for your wonderful site. I have so much material to use in class and inspire me to try something a little different more often. I am going to show my maths department your website and encourage them to use it too. How lovely that you have compiled such a great resource to help teachers and pupils.
Thanks again"

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Featured Activity

Digital Darts

Digital Darts

An online darts game for one or two players requiring skill, strategy and mental arithmetic. It provides an enjoyable way to practise adding and subtracting two and three digit numbers.


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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


Two unknowns - You really should start here before taking on three unknowns.

Level 1 - Standard set of questions with the equations set out in a familiar way

Level 2 - A mixed up collection of equations to challenge the high achiever

Level 3 - An awful heap of tedious equations generated by AI where the solutions are vulgar fractions (not recommended)

More on this topic including lesson Starters, visual aids, investigations and self-marking exercises.

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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The video above is from Corbett Maths.

Example using matrix row reduction:

Solve the following system of equations:

$$x+y+z=3$$ $$2x-y+z=0$$ $$x-2y-z=-3$$

We begin by writing the augmented matrix for the system of equations:

$$\left[\begin{array}{ccc|c}1 & 1 & 1 & 3\\2 & -1 & 1 & 0\\1 & -2 & -1 & -3\end{array}\right]$$

We now perform row operations to transform this matrix into echelon form:

\(R2 = R2 - 2 \times R1\)

$$\left[\begin{array}{ccc|c}1 & 1 & 1 & 3\\0 & -3 & -1 & -6\\1 & -2 & -1 & -3\end{array}\right]$$

\(R3 = R3 - R1\)

$$\left[\begin{array}{ccc|c}1 & 1 & 1 & 3\\0 & -3 & -1 & -6\\0 & -3 & -2 & -6\end{array}\right]$$

\(R3 = R3 - R2\)

$$\left[\begin{array}{ccc|c}1 & 1 & 1 & 3\\0 & -3 & -1 & -6\\0 & 0 & -1 & 0\end{array}\right]$$

The augmented matrix is now in the form:

$$\left[\begin{array}{ccc|c}a & b & c & d\\0 & e & f & g\\0 & 0 & h & i\end{array}\right]$$

If \( h \neq 0\) there is a unique solution.

If \( h = 0 \text{ and } i \neq 0 \) there is no solution

If \( h = 0 \text{ and } i = 0 \) there are infinitely many solutions (let \(z=t\)).

In the example above ...

from \(R3\) it can be seen that \( z=0 \)

from \(R2\) it can be seen that \( y=2 \)

from \(R1\) it can be seen that \( x=1 \)

The solutions are \( x=1,y=2,z=0 \).

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