How do solve the following linear system?:  -4x-2y=14 , -10x+7y=-2 ?

Jan 24, 2018

See a solution process below:

Explanation:

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

$- 4 x - 2 y = 14$

$\textcolor{red}{- \frac{1}{2}} \left(- 4 x - 2 y\right) = \textcolor{red}{- \frac{1}{2}} \times 14$

$\left(\textcolor{red}{- \frac{1}{2}} \times - 4 x\right) + \left(\textcolor{red}{- \frac{1}{2}} \times - 2 y\right) = - 7$

$2 x + 1 y = - 7$

$2 x + y = - 7$

$2 x - \textcolor{red}{2 x} + y = - 7 - \textcolor{red}{2 x}$

$0 + y = - 7 - 2 x$

$y = - 7 - 2 x$

Step 2) Substitute $\left(- 7 - 2 x\right)$ for $y$ in the second equation and solve for $x$:

$- 10 x + 7 y = - 2$ becomes:

$- 10 x + 7 \left(- 7 - 2 x\right) = - 2$

$- 10 x + \left(7 \times - 7\right) + \left(7 \times - 2 x\right) = - 2$

$- 10 x + \left(- 49\right) + \left(- 14 x\right) = - 2$

$- 10 x - 49 - 14 x = - 2$

$- 10 x - 14 x - 49 = - 2$

$\left(- 10 - 14\right) x - 49 = - 2$

$- 24 x - 49 = - 2$

$- 24 x - 49 + \textcolor{red}{49} = - 2 + \textcolor{red}{49}$

$- 24 x - 0 = 47$

$- 24 x = 47$

$\frac{- 24 x}{\textcolor{red}{- 24}} = \frac{47}{\textcolor{red}{- 24}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{- 24}}} x}{\cancel{\textcolor{red}{- 24}}} = - \frac{47}{24}$

$x = - \frac{47}{24}$

Step 3) Substitute $- \frac{47}{24}$ for $x$ in the solution to the first equation at the end of Step 1:

$y = - 7 - 2 x$ becomes:

$y = - 7 - \left(2 \times - \frac{47}{24}\right)$

$y = - 7 - \left(- \frac{47}{12}\right)$

$y = - 7 + \frac{47}{12}$

$y = \left(\frac{12}{12} \times - 7\right) + \frac{47}{12}$

$y = - \frac{84}{12} + \frac{47}{12}$

$y = - \frac{37}{12}$

The Solution Is:

$x = - \frac{47}{24}$ and $y = - \frac{37}{12}$

Or

$\left(- \frac{47}{24} , - \frac{37}{12}\right)$