# How do you solve 2x+4y=6 and 4x-3y=-10?

Jun 18, 2018

$x = - 1 , y = 2$

#### Explanation:

Given
[1]$\textcolor{w h i t e}{\text{XXX}} 2 x + 4 y = 6$
[2]$\textcolor{w h i t e}{\text{XXX}} 4 x - 3 y = - 10$

Method 1: By elimination
We note that if we multiply equation [1] by $2$ the coefficient of $x$ will be the same as that of equation [2]
[3]$\textcolor{w h i t e}{\text{XXX}} 4 x + 8 y = 12$
We can now eliminate the $x$ term by subtracting equation [2] from equation [3]
$\textcolor{w h i t e}{\text{XXX.."[x])4x+8y=color(white)("x..}} 12$
$\textcolor{w h i t e}{\text{XXX}} - \left(\underline{4 x - 3 y = - 10}\right)$
[4]$\textcolor{w h i t e}{\text{XXXXXx..")11y=color(white)("x..}} 22$
After dividing both sides by $11$
[5]$\textcolor{w h i t e}{\text{XXX}} y = 2$
Then substituting $2$ for $y$ in [1]
[6]$\textcolor{w h i t e}{\text{XXX}} 2 x + 4 \cdot 2 = 6$
which simplifies as
[7]$\textcolor{w h i t e}{\text{XXX}} 2 x = - 2$
or
[8]$\textcolor{w h i t e}{\text{XXX}} x = - 1$

Method 2: By substitution
We note that we can divide both sides of [1] by $2$ and then re-arrange the terms to get
[9]$\textcolor{w h i t e}{\text{XXX}} x = 3 - 2 y$
Now we can substitute $\left(3 - 2 y\right)$ for $x$ in [2]
[10]$\textcolor{w h i t e}{\text{XXX}} 4 \cdot \left(3 - 2 y\right) - 3 y = - 10$
Simplifying [10]
[11]$\textcolor{w h i t e}{\text{XXX}} 12 - 8 y - 3 y = - 10$
then
[12]$\textcolor{w h i t e}{\text{XXX}} - 11 y = - 22$
and finally (after dividing both sides by $2$
[13]$\textcolor{w h i t e}{\text{XXX}} y = 2$
Substituting $\left(2\right)$ for $y$ back in [1] (or [9]) gives
[14}$\textcolor{w h i t e}{\text{XXX}} x = - 1$