# Thread: simultaneous equations containing fractions

1. ## simultaneous equations containing fractions

Hi;
I have these equations 5/x - 2/y = 2 and 2/x + 3/y = 16.

Unfortunatly I have no work to show because I don't know where to start with unknows on the bottom.

Thanks

2. ## Re: simultaneous equations containing fractions

Do you have to solve a system? Or are the two equations totally independent of each other? (I think the first one)
In case you mean a sytem it can be useful to do a substitution, let $\frac{1}{x}=a$ and let $\frac{1}{y}=b$ (and $x,y \neq 0$), solve the system for a and b and afterwards do the back-substitution.

3. ## Re: simultaneous equations containing fractions

Originally Posted by anthonye
Hi;
I have these equations 5/x - 2/y = 2 and 2/x + 3/y = 16.

Unfortunatly I have no work to show because I don't know where to start with unknows on the bottom.

Thanks
$x\neq 0$ and $y\neq 0$.

Substitute:

$\frac{1}{x}=t$ and $\frac{1}{y}=u$.

4. ## Re: simultaneous equations containing fractions

Originally Posted by anthonye
Hi;
I have these equations 5/x - 2/y = 2 and 2/x + 3/y = 16.
Here is a start.
$\left. \begin{gathered} \frac{5}{x} - \frac{2}{y} = 2 \hfill \\ \frac{2}{x} + \frac{3}{y} = 16 \hfill \\ \end{gathered} \right\} \equiv \left\{ \begin{gathered} \frac{{15}}{x} - \frac{6}{y} = 6 \hfill \\ \frac{4}{x} + \frac{6}{y} = 32 \hfill \\ \end{gathered} \right.$

5. ## Re: simultaneous equations containing fractions

Plato my result from here is 0=0 why.

6. ## Re: simultaneous equations containing fractions

Originally Posted by anthonye
Plato my result from here is 0=0 why.
What do you mean with that?
$x=y\neq 0$, because else you're diving by zero which is offcourse not possible (undefined).

EDIT:
If you use 'Plato's method' then make the sum of the two equations in the system therefore you'll get an equation in one variable which you can solve directly.

7. ## Re: simultaneous equations containing fractions

Do I need to flip the fractions to solve them?

8. ## Re: simultaneous equations containing fractions

Originally Posted by anthonye
Do I need to flip the fractions to solve them?
Make the sum of the two equations in the system, see my EDIT in my previous post.

9. ## Re: simultaneous equations containing fractions

Originally Posted by anthonye
Do I need to flip the fractions to solve them?
Add these together $\left\{ \begin{gathered} \frac{{15}}{x} - \frac{6}{y} = 6 \hfill \\ \frac{4}{x} + \frac{6}{y} = 32 \hfill \\ \end{gathered} \right.$

to get $\frac{{19}}{x} = 38$. Surely you can solve that.

So x = 0.5

11. ## Re: simultaneous equations containing fractions

Originally Posted by anthonye
So x = 0.5
That's correct, substitute $x=0,5$ in one of the equations and solve the equation to $y$.

12. ## Re: simultaneous equations containing fractions

Ok thanks for the help just one more thing can they be solved by flipping them?

13. ## Re: simultaneous equations containing fractions

Then you've to make a common denominator, if you want to flipp them then you've to write:
$\frac{5}{x}-\frac{2}{y}=2 \Leftrightarrow \frac{5y-2x}{xy}=2 \Leftrightarrow \frac{xy}{5y-2x}=\frac{1}{2}$
You see that this is not an easier way.

14. ## Re: simultaneous equations containing fractions

Ok I'll look more later thanks for now.

15. ## Re: simultaneous equations containing fractions

I will give you my solution:
Let $\frac{1}{x}=a$ and let $\frac{1}{y}=b$ therefore you can write the equation as:
$\left\{ \begin{gathered} 5a - 2b = 2 \hfill \\ 2a +3b = 16 \hfill \\ \end{gathered} \right.$
$\Leftrightarrow \left\{ \begin{gathered} 15a - 6b = 6 \hfill \\ 4a +6b = 32 \hfill \\ \end{gathered} \right.$
$\Leftrightarrow 19a=38 \Leftrightarrow a=2$

If we substitute $a=2$ in one of the equations:
$10-2b=2 \Leftrightarrow 2b=8 \Leftrightarrow b=4$

Back-substitution:
If $a=2 \ \mbox{and} \ \frac{1}{x}=a \Rightarrow x=\frac{1}{2}$
$y=\frac{1}{4}$

You can check your solutions by entering them into the sytem.

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