# 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.

 1 $$2x+y-3z= -4 \\ 3x+y+z= 15 \\ x-y+2z = 9$$ $$x =$$ ✓ ✗ ☐ $$y =$$ ✓ ✗ ☐ $$z =$$ ✓ ✗ 2 $$5a+2b+c=18 \\ 3a+4b+2c= 22 \\ a+5b+c=19$$ $$a =$$ ✓ ✗ ☐ $$b =$$ ✓ ✗ ☐ $$c =$$ ✓ ✗ 3 $$2d+3e-4f = -6 \\ d-e-f= -7\\ 9d+2e-2f=23$$ $$d =$$ ✓ ✗ ☐ $$e =$$ ✓ ✗ ☐ $$f =$$ ✓ ✗ 4 $$g-7h-7i=-42 \\ 2g-2h+i= 12\\ 5g+3h+i = 48$$ $$g =$$ ✓ ✗ ☐ $$h =$$ ✓ ✗ ☐ $$i =$$ ✓ ✗ 5 $$j+k+l= 8 \\ 2j-3k+9l= 34\\ -j+k-3l=-12$$ $$j =$$ ✓ ✗ ☐ $$k =$$ ✓ ✗ ☐ $$l =$$ ✓ ✗ 6 $$-m+4n-p=-2 \\ 2m-3n+5p= 21\\ 7m-n+p=17$$ $$m =$$ ✓ ✗ ☐ $$n =$$ ✓ ✗ ☐ $$p =$$ ✓ ✗ 7 $$2q+2r-5s=-53 \\ q-r+2s= 14\\ -q+5r+5s=-15$$ $$q =$$ ✓ ✗ ☐ $$r =$$ ✓ ✗ ☐ $$s =$$ ✓ ✗ 8 $$t-5u-5v=43 \\ 2t+u+3v= -19\\ -5t+5u-v=-17$$ $$t =$$ ✓ ✗ ☐ $$u =$$ ✓ ✗ ☐ $$v =$$ ✓ ✗ 9 $$9w+x-y=78 \\ 2w+7x-3y= 25\\ -7w+5x+5y=-8$$ $$w =$$ ✓ ✗ ☐ $$x =$$ ✓ ✗ ☐ $$y =$$ ✓ ✗ 10 $$8x-5y+2z=33 \\ -3x+7y-z= -58\\ 2z-5x-y=10$$ $$x =$$ ✓ ✗ ☐ $$y =$$ ✓ ✗ ☐ $$z =$$ ✓ ✗
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This is Three Unknowns level 1. You can also try:
Level 2 Level 3

## Instructions

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.

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#### Striped Sweets

Colour the sweet wrappers so that no two are the same. A multi-level activity designed to encourage a systematic strategy for finding all of the different permutations.

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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.

## Teachers

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## Description of Levels

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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.

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## Help

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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