# How do solve the following linear system?:  8x+2y=3 , 2x+7=-5y ?

Jun 27, 2018

$8 x + 2 y = 3$

$2 x + 7 = - 5 y$

$\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$

Let's isolate for $y$ in the second equation:

$\frac{- 2 x - 7}{5} = y$

Now plug that into the first equation for $y$:

$8 x + \frac{2 \left(- 2 x - 7\right)}{5} = 3$

$\textcolor{g r a y}{\frac{5}{5}} \times 8 x - \frac{4 x}{5} - \frac{14}{5} = 3$

common denominator

$\frac{40 x - 4 x - 14}{5} = 3$

multiply both sides by $5$

$36 x = 29$

color(green)(x = 29/36 = 0.806

$\textcolor{w h i t e}{. .}$

Now let's solve for $y$:

$8 \left(\frac{29}{36}\right) + 2 y = 3 \times \textcolor{g r a y}{\frac{36}{36}}$

$\frac{232}{36} + 2 y = \frac{108}{36}$

$2 y = \frac{108}{36} - \frac{232}{36}$

$2 y = - \frac{124}{36}$

$y = - \frac{124}{72} = - 1.722$

$\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$

Let's check our work by graphing the two equations and seeing where they intersect

Looks right! Good job

Jun 27, 2018

$x = \frac{29}{36} , \textcolor{w h i t e}{\text{xx}} y = - \frac{31}{18}$

#### Explanation:

Given
[1]$\textcolor{w h i t e}{\text{XXX}} 8 x + 2 y = 3$
[2]$\textcolor{w h i t e}{\text{XXX}} 2 x + 7 = - 5 y$

Converting [2] into standard form:
[3]$\textcolor{w h i t e}{\text{XXX}} 2 x + 5 y = - 7$

We note that if we multiply [3] by $4$ the coefficient of $x$ becomes the same as that of [1]
[4]$\textcolor{w h i t e}{\text{XXX}} 8 x + 20 y = - 28$

Subtracting [1] from [4] (to get rid of the $x$ variable
[5]$\textcolor{w h i t e}{\text{XXX}} 18 y = - 31$

Dividing both sides of [5] by $18$
[6]$\textcolor{w h i t e}{\text{XXX}} y = - \frac{31}{18}$

Substituting $\left(- \frac{31}{8}\right)$ for $y$ in [1]
[7]$\textcolor{w h i t e}{\text{XXX}} 8 x + 2 \cdot \left(- \frac{31}{18}\right) = 3$

[8]$\textcolor{w h i t e}{\text{XXX")8xcolor(white)("xxxxxxxxxxxx}} = 3 + \frac{31}{9} = \frac{58}{9}$

[9]$\textcolor{w h i t e}{\text{XXX}} x = \frac{29}{36}$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Verifying by substituting $\left(- \frac{31}{18}\right)$ for $y$ and $\frac{29}{36}$ for $x$ in [2]
[10]$\textcolor{w h i t e}{\text{XXX")2 * (29/36)+7 color(white)("xxx")?=?color(white)("xxx}} - 5 \cdot \left(- \frac{31}{18}\right)$

[11]$\textcolor{w h i t e}{\text{XXX")29/18+126/18color(white)("xxxxx")?=?color(white)("xx}} \frac{155}{18}$

Yes: results verified!