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Diophantine Equations

Practise finding integer solutions to equations with more than one unknown.
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This is level 4: non linear diophantine equations. You will be awarded a trophy if you get at least 7 answers correct and you do this activity online.

Find a set of positive whole number solutions to the following equations, though there may be more.

1

$$\dfrac{x}{2y} = \dfrac{3}{4}$$

\(x=\) \( \quad y=\) ☐ ✓ ✗

2

$$w^2-20=80-z^2$$

\(w=\) \( \quad z=\) ☐ ✓ ✗

3

$$a^2=b^2+c^2$$

\(a=\) \( \quad b=\)

\( \quad c=\) ☐ ✓ ✗

4

$$\sqrt{m} + \sqrt{n} = 10$$

\(m=\) \( \quad n=\) ☐ ✓ ✗

5

$$x^y=256$$

\(x=\) \( \quad y=\) ☐ ✓ ✗

6

$$\dfrac{j}{5} + \dfrac{k}{15} = 6$$

\(j=\) \( \quad k=\) ☐ ✓ ✗

7

$$3^c \times 9^d=6561$$

\(c=\) \( \quad d=\) ☐ ✓ ✗

8

$$3^a+9^b=1458$$

\(a=\) \( \quad b=\) ☐ ✓ ✗

9

$$5x+7y−9z=13$$

\(x=\) \( \quad y=\)

\( \quad z=\) ☐ ✓ ✗

Check

This is Diophantine Equations level 4. You can also try:
Level 1 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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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 26 March 'Starter of the Day' page by Julie Reakes, The English College, Dubai:

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"My year 8's absolutely loved the "Separated Twins" starter. I set it as an optional piece of work for my year 11's over a weekend and one girl came up with 3 independant solutions."

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Description of Levels

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Level 1 - Linear Diophantine equations of the form \(ax+by=c\)

Level 2 - Equations derived from real life situations

Level 3 - Similar to Level 1 but with larger numbers

Level 4 - Non linear Diophantine equations

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

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Example

A classic example of a Diophantine equation is \(3x + 2y = 11\).
The goal is to find values for \(x\) and \(y\) that are whole numbers (integers) and satisfy this equation.

To solve this, we can start by trying different values for \(x\) and see if we can find a corresponding value for \(y\) that makes the equation true. For example, let's try \(x = 1\):

\(3(1) + 2y = 11\)
\(3 + 2y = 11\)
\(2y = 11 - 3\)
\(2y = 8\)
\(y = \frac{8}{2}\)
\(y = 4\)

So, when \(x = 1\), \(y = 4\). Therefore, one solution to the equation is \(x = 1\) and \(y = 4\).

We can check if there are other solutions by trying different values for \(x\). If we try \(x = 2\):

\(3(2) + 2y = 11\)
\(6 + 2y = 11\)
\(2y = 11 - 6\)
\(2y = 5\)

This doesn't give us an integer solution for \(y\), since 5 divided by 2 is 2.5, not a whole number. Therefore, \(x = 2\) doesn't work.

Similarly, if we try \(x = 3\):

\(3(3) + 2y = 11\)
\(9 + 2y = 11\)
\(2y = 11 - 9\)
\(2y = 2\)
\(y = \frac{2}{2}\)
\(y = 1\)

So, another solution is \(x = 3\) and \(y = 1\).

Therefore, the Diophantine equation \(3x + 2y = 11\) has two positive integer solutions:
\((x, y) = (1, 4)\) and \((x, y) = (3, 1)\).

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.

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