# How do you solve the following linear system:  -2x+5y=4, 3x+8y=1 ?

May 20, 2016

(x,y)=color(blue)(""(-27/31,14/31))

#### Explanation:

Given
[1]$\textcolor{w h i t e}{\text{XXX}} - 2 x + 5 y = 4$
[2]$\textcolor{w h i t e}{\text{XXX}} 3 x + 8 y = 1$

To eliminate the $x$ components we can convert the given equations
into ones with identical coefficients for $x$
by multiplying [1] by $3$ and [2] by $2$
[3]$\textcolor{w h i t e}{\text{XXX}} - 6 x + 15 y = 12$
[4]$\textcolor{w h i t e}{\text{XXX}} 6 x + 16 y = 2$

[5]$\textcolor{w h i t e}{\text{XXX")31y=14color(white)("XX")rarrcolor(white)("XX}} y = \frac{14}{31}$
We cold plug this value back into one of the original equations and solve for $x$
or (the method I find simpler in this case) repeat the above process to eliminate $y$ from the original given equations.
[6]$\textcolor{w h i t e}{\text{XXX}} - 16 x + 40 y = 32$
[7]$\textcolor{w h i t e}{\text{XXX}} 15 x + 40 y = 5$
[8]$\textcolor{w h i t e}{\text{XXX")-31x = 27color(white)("XX")rarrcolor(white)("XX}} x = - \frac{27}{31}$