How do you solve the following system?: 5x -7y =-43 , -13x +y = 8

Jun 3, 2017

See a solution process below:

Explanation:

Step 1) Solve the second equation for $y$:

$- 13 x + y = 8$

$\textcolor{red}{13 x} - 13 x + y = \textcolor{red}{13 x} + 8$

$0 + y = 13 x + 8$

$y = 13 x + 8$

Step 2) Substitute $\left(13 x + 8\right)$ for $y$ in the first equation and solve for $x$:

$5 x - 7 y = - 43$ becomes:

$5 x - 7 \left(13 x + 8\right) = - 43$

$5 x - \left(7 \cdot 13 x\right) - \left(7 \cdot 8\right) = - 43$

$5 x - 91 x - 56 = - 43$

$\left(5 - 91\right) x - 56 = - 43$

$- 86 x - 56 = - 43$

$- 86 x - 56 + \textcolor{red}{56} = - 43 + \textcolor{red}{56}$

$- 86 x - 0 = 13$

$- 86 x = 13$

$\frac{- 86 x}{\textcolor{red}{- 86}} = \frac{13}{\textcolor{red}{- 86}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{- 86}}} x}{\cancel{\textcolor{red}{- 86}}} = - \frac{13}{86}$

$x = - \frac{13}{86}$

Step 3) Substitute $- \frac{13}{86}$ for $x$ in the solution to the second equation at the end of Step 1 and calculate $y$;

$y = 13 x + 8$ becomes:

$y = \left(13 \cdot - \frac{13}{86}\right) + 8$

$y = - \frac{169}{86} + 8$

$y = - \frac{169}{86} + \left(\frac{86}{86} \times 8\right)$

$y = - \frac{169}{86} + \frac{688}{86}$

$y = \frac{519}{86}$

The solution is: $x = - \frac{13}{86}$ and $y = \frac{519}{86}$ or $\left(- \frac{13}{86} , \frac{519}{86}\right)$