# How do solve the following linear system?:  11x + 3y + 7 = 0 , -6x-2y=-8 ?

Jun 20, 2018

$x = - \frac{19}{2} , y = \frac{65}{2}$

#### Explanation:

Multiplying the first equation by $2$ and the second equation by $3$ and adding both

$4 x = - 38$
so
$x = - \frac{19}{2}$
plugging this in the second equation
$- 6 \left(- \frac{19}{2}\right) - 2 y = - 8$
$57 - 2 y = - 8$
$- 2 y = - 65$

$y = \frac{65}{2}$

Jun 20, 2018

$x = - \frac{19}{2}$ and $y = \frac{65}{2}$

#### Explanation:

Given
[1]$\textcolor{w h i t e}{\text{XXX}} 11 x + 3 y + 7 = 0$
[2]$\textcolor{w h i t e}{\text{XXX}} - 6 x - 2 y = - 8$

To make this easier to work with I will convert [1] into the same standard form as [2] by subtracting $7$ from both sides
[3]$\textcolor{w h i t e}{\text{XXX}} 11 x + 3 y = - 7$

We need to eliminate one of the variables (either $x$ or $y$) by combining these two equations.
For example, we might decide to eliminate the term containing the variable $y$.

To do this we need to convert equations [2] and [3] into equivalent forms with identical magnitude coefficients for $y$.

The least common multiple for the (magnitude of)coefficients of $y$ in [2] and [3] is $6$, so we will convert each of [2] and [3] into equivalent forms with a term $\pm 6 y$.

Multiplying [2] by $3$
[4]$\textcolor{w h i t e}{\text{XXX}} - 18 x - 6 y = - 24$
Multiplying [3] by $2$
[5]$\textcolor{w h i t e}{\text{XXX}} 22 x + 6 y = - 14$

Now if we add [4] and [5], we can eliminate the $y$ term:
[6]$\textcolor{w h i t e}{\text{XXX}} 4 x = - 38$

Dividing both sides of [6] by $4$ we get
[7]$\textcolor{w h i t e}{\text{XXX}} x = - \frac{19}{2}$

We can now substitute $\left(- \frac{19}{2}\right)$ for $x$ back in one of our original equations. For demonstration purposes I will use equation [2]
[8]$\textcolor{w h i t e}{\text{XXX}} - 6 \cdot \left(- \frac{19}{2}\right) - 2 y = - 8$

Simplifying
[9]$\textcolor{w h i t e}{\text{XXX}} 57 - 2 y = - 8$

[10]$\textcolor{w h i t e}{\text{XXX}} - 2 y = - 65$

[11]$\textcolor{w h i t e}{\text{XXX}} y = \frac{65}{2}$

Giving the final solution
color(white)("XXX")x=-19/2 " or " -9 1/2
and
color(white)("XXX")y=65/2" or " 32 1/2

[Since we used [2] to develop the value for $y$ we should check this answer using [1] to verify that our results are correct].