# How do you solve x + 2y = 6 and x - 4y = 8 using substitution?

Apr 10, 2016

If $x - 4 y = 8$
then $x = 4 y + 8$

Therefore we can substitute $\textcolor{b l u e}{4 y + 8}$ for $x$
in the other given equation: $x + 2 y = 6$
giving:
$\textcolor{w h i t e}{\text{XXX}} \textcolor{b l u e}{4 y + 8} + 2 y = 6$
which simplifies as:
$\textcolor{w h i t e}{\text{XXX}} 6 y = - 2$
or
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{red}{- \frac{1}{3}}$

We can then substitute $\textcolor{red}{- \frac{1}{3}}$ for $y$
in one of the given equations, say: $x - 4 y = 8$
giving:
$\textcolor{w h i t e}{\text{XXX}} x - 4 \left(- \frac{1}{3}\right) = 8$

$\textcolor{w h i t e}{\text{XXX}} x + \frac{4}{3} = 8$

$\textcolor{w h i t e}{\text{XXX}} x = 6 \frac{2}{3}$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Verification is always a good idea.
So testing the above solution $\left(x , y\right) = \left(6 \frac{2}{3} , - \frac{1}{3}\right)$ in the given equation not used to evaluate $x$
$\textcolor{w h i t e}{\text{XXX}} x + 2 y = 6$ (????)

$\textcolor{w h i t e}{\text{XXX}} 6 \frac{2}{3} + 2 \times \left(- \frac{1}{3}\right) = 6$ (????)

$\textcolor{w h i t e}{\text{XXX}} 6 \frac{2}{3} - \frac{2}{3} = 6$ (Correct!)

Apr 10, 2016

substitute one of the following equations

#### Explanation:

x+2y=6 -----------------(1) and x-4y=8------------------(2)

Steps:
~>. Substitute one of the following equations, either (1) OR (2)

substituting (1)

Put x subject of formula,

x= 6-2y--------------(*)

~> Replace the equation (*) in equation (2)

x-4y=8
replacing.....

(6-2y)-4y=8
6-2y-4y=8
6-6y=8

Transpose "+6"on the other side of the equation
-6y=8-6
-6y=2

y= $\frac{2}{-} 6$

y=$- \left(\frac{1}{3}\right)$

Hence, Replace the value obtained for y in equation (*)

x=6-2y
x=6-($2 \cdot - \frac{1}{3}$)
x=6-($- \frac{2}{3}$)
x=6+($\frac{2}{3}$)
x=$\frac{20}{3}$

Therefore: x=$\frac{20}{3}$ and y=$- \left(\frac{1}{3}\right)$