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I am sorry if this is too easy for you, but it's been very many decades since I had math at school. ;-) Please also consider that English is a foreign language for me.

I need to resolve X and Y for this equation system:

\begin{align} x\cdot y &= 6930\\ \frac yx &= 1.41 \end{align}

How can this be done and which are the steps to the solution? Please write in easy to understand arithmetic language (no advanced mathematical symbols please).

Many thanks in advance!

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    I'll leave this as a comment on the offchance that $1.41$ is an "engineering approximation" for $\sqrt{2}$ but exercising that approximation too soon loses a full significant digit: $\sqrt{\frac{6930}{1.41}} \simeq 70.106$ vs. $\sqrt{\frac{6930}{\sqrt{2}}} \simeq 70.001$. – 2017-01-09
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    Thanks, 1.41 in my specific case is a "good enough" approximation for the calculation of a rectangle model where starting from the aspect ratio of the edges of a DIN page (1.41) the edges of the rectangle change and the area of the rectangle must remain the same (6930). The solution I received here helped me to achieve this. – 2017-01-09
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    BTW, there is an error in the changed title: The second equation is not "x/y = 1.41" but "y/x = 1.41". Sorry, I don't know how the mathematical formatting is done. – 2017-01-09

4 Answers 4

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First divide the first equation by X:

$$Y = 6930/X$$

Then substitute the Y into the 2nd equation.

$$(6930/X)/X=1.41$$

Multiply by X:

$$6930/X = 1.41X$$

... again

$$6930=1.41X^2$$

Divide by 1.41

$$4914.89=X^2$$

Take square root:

$$X= +/-70.10$$

Solve the Y by yourself :).

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If $y/x = 1.41$, then $y = 1.41x$. So plugging this in the first equation yields $1.41x^2 = 6930$ and thus $x = \pm \sqrt{6930/1.41}$. Now it should be easy to find also $y$.

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The first equation $$ x \cdot y = 6930 \quad (*) $$ implies that both $x$ and $y$ can not be zero.

So the second equation $$ \frac{y}{x} = 1.41 \quad (**) $$ is equivalent (has the same solutions) to $$ y = 1.41 \cdot x $$ We can insert this into the first equation and get $$ x \cdot (1.41 \cdot x) = 6930 $$ which simplifies to $$ x^2 = \frac{6930}{1.41} $$ or $$ x = \pm \sqrt{\frac{6930}{1.41}} $$ where the $\pm$ is short hand notation for two solutions, one positive, one negative.

Finally we get $$ y = 1.41 \cdot x = 1.41 \cdot \pm \sqrt{\frac{6930}{1.41}} = \pm \sqrt{1.41 \cdot 6930} $$ So we got four solutions in total, however equation $(*)$ or $(**)$ will only permit pairs of same signs, as we need a positive product or fraction, so we end up with two solutions: $$ (x, y) = \pm \left( \sqrt{\frac{6930}{1.41}}, \sqrt{1.41 \cdot 6930}\right) $$

Update:

Here you can fiddle with GeoGebra: link

This free software allows you to solve the problem graphically, symbolic and numeric.

enter image description here

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    Thanks to everybody, you helped me a lot! Now I understand the problem better. – 2017-01-08
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    BTW, is there a (free) software program (Windows 7) where I can put in these two equations in text form and it solves it? – 2017-01-08
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    @user1580348: You can use the Wolfram Alpha website: http://www.wolframalpha.com/input/?i=xy+%3D+6930,+y%2Fx+%3D+1.41 – 2017-01-08
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    @user1580348 Would second Adreas' recommendation of Wolfram Alpha, however if you're specifically looking for local software (that works offline) I can recommend the "Equation Solver" feature of Microsoft Mathematics (Availible here: https://www.microsoft.com/en-gb/download/details.aspx?id=15702) – 2017-01-08
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    Thanks again to everybody! It is wonderful that there is a community where knowledge can be shared! Hopefully one day I can share some bit of knowledge too! – 2017-01-08
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    I added a link to a GeoGebra work sheet. It allows to solve the problem graphically, just enter the equations and try the intersect tool Or you can enter it in a CAS (Computer Algebra System) window and solve it symbolically or numerically. – 2017-01-08
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    Thank you, this looks really beautiful! I think I start to like Mathematics! – 2017-01-09
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Start with$$\begin{align*} & xy=6930\tag1\\ & \tfrac yx=1.41\tag2\end{align*}$$ Multiplying the numerator and denominator of $(2)$ by $y$ gives$$\begin{align*} & \dfrac {y^2}{xy}=1.41\tag3\\ & y^2=1.41\cdot xy\tag4\\ & y^2=1.41\cdot 6930\tag5\\ & y=\pm\sqrt{9771.3}\end{align*}$$ Substitute that back into $(1)$ to get the corresponding $x$ values.