# Thread: Puzzle: 4 Simultaneous Equastions

1. ## Puzzle: 4 Simultaneous Equastions

I am new, and not sure if this is the right forum. Here is my puzzle

Givenx1y1a11+x1y2a12=R1,
x2y1a21+x2y2a22=R2,
x1y1a11+x2y1a21=C1 and
x1y2a12+x2y2a22=C2,

can x1 be expressed in terms that do not include x2, y1 or y2?

2. ## Re: Puzzle: 4 Simultaneous Equastions

Are you serious?!

You use a11, a12, a21, a22 as variables:
why not a1, a2, a3, a4 instead?
At least, that mess would be more readable...

I hope someone else here will help...won't be me!!

3. ## Re: Puzzle: 4 Simultaneous Equastions

Originally Posted by MinimalBob
I am new, and not sure if this is the right forum. Here is my puzzle

Givenx1y1a11+x1y2a12=R1,
x2y1a21+x2y2a22=R2,
x1y1a11+x2y1a21=C1 and
x1y2a12+x2y2a22=C2,

can x1 be expressed in terms that do not include x2, y1 or y2?
Yes, but there may be more than one solution.

Solve the third and forth equations for $y_1$ and $y_2$ respectively, plugging the results into the first two equations.
Let
$\alpha = a_{21}a_{22}(C_1+C_2-R_2)$
$\beta = a_{12}a_{21}(C_1-R_2)+a_{11}a_{22}(C_2-R_2)$
$\gamma = -a_{11}a_{21}R_2$

Then, you have $\alpha x_2^2+\beta x_1x_2+\gamma x_1^2 = 0$

Then $x_2 = x_1 \left( \dfrac{-\beta \pm \sqrt{\beta^2-4\alpha\gamma}}{2\alpha} \right)$

So, let $r_1 = \dfrac{-\beta+\sqrt{\beta^2-4\alpha\gamma}}{2\alpha}$ and $r_2 = \dfrac{-\beta-\sqrt{\beta^2-4\alpha\gamma}}{2\alpha}$ (Note: $r_1$ and $r_2$ do not depend on any of the variables you mentioned are not valid for dependencies).

Then, you have $x_2 = r_1x_1$ or $x_2 = r_2x_1$.

Plug this in to the first equation, and you will get a cubic or quartic polynomial for x_1 in terms of $a_{11},a_{12},a_{21},a_{22},C_1,C_2,R_1,R_2$. For help solving cubic or quartic polynomials, that is another problem altogether.