How do you solve 8x = -2y - 10 and 2x = 4y using substitution?

May 10, 2016

(x,y)=color(blue)(""(-10/9,-5/9))

Explanation:

Given:
[1]$\textcolor{w h i t e}{\text{XXX}} 8 x = - 2 y - 10$
[2]$\textcolor{w h i t e}{\text{XXX}} 2 x = 4 y$

Dividing both sides of [2] by 2
[3]$\textcolor{w h i t e}{\text{XXX}} x = 2 y$

From [3] we see that we can substitute $2 y$ anywhere for $x$
and specifically we can substitute $2 y$ for $x$ in [1] to get
[4]$\textcolor{w h i t e}{\text{XXX}} 8 \left(2 y\right) = - 2 y - 10$

Simplifying:
[5]$\textcolor{w h i t e}{\text{XXX}} 16 y = - 2 y - 10$

[6]$\textcolor{w h i t e}{\text{XXX}} 18 y = - 10$

[7]$\textcolor{w h i t e}{\text{XXX}} y = - \frac{5}{9}$

We can now substitute $\left(- \frac{5}{9}\right)$ for $y$ in [3] to get
[8]$\textcolor{w h i t e}{\text{XXX}} x = 2 \left(- \frac{5}{9}\right) = - \frac{10}{9}$