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

Jun 10, 2018

$x = - \frac{12}{17} , y = - \frac{4}{17}$

Explanation:

Multiplying the first equation by $3$ and the second one by $4$ and adding both we get
$- 17 y = 4$
so

$y = - \frac{4}{17}$
now we can calculate $x$
$4 x + 5 \left(- \frac{4}{17}\right) = - 4$
so

$x = \frac{1}{4} \cdot \left(- 4 + \frac{20}{17}\right)$

so

$x = - \frac{12}{17}$

Jun 10, 2018

$x = - \frac{12}{17} \textcolor{w h i t e}{\text{dddd}} y = - \frac{4}{17}$

Explanation:

Given:

$4 x + 5 y = - 4 \text{ } \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots E q u a t i o n \left(1\right)$
$- 3 x - 8 y = 4 \text{ } \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . E q u a t i o n \left(2\right)$

2 equations and 2 unknowns. Thus solvable.

We need to end up with one equation and 1 unknown.

$\textcolor{b l u e}{\text{'Getting rid' of the "x" term}}$

$\left[\textcolor{w h i t e}{\text{d")3xxEqn(1)color(white)(2/2)]+[color(white)("d}} 4 \times E q n \left(2\right) \textcolor{w h i t e}{\frac{2}{2}}\right]$

color(white)("d.")12x+15y=-12" "........................Equation(1_a)
ul(-12x-32y=color(white)("d..")16)" "........................Equation(2_a)
$\textcolor{w h i t e}{\text{dd")0color(white)(".d}} - 17 y = + 4$

Divide both sides by $- 17$

$\textcolor{w h i t e}{\text{dddddddd}} y = - \frac{4}{17}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{'Getting rid' of the "y" term}}$

Substitute $- \frac{4}{17}$ for $y$

I choose $E q n \left(1\right)$ as all the left side of = is positive.
It will still work if you choose $E q n \left(2\right)$

$\textcolor{g r e e n}{4 x + 5 \textcolor{red}{y} = - 4 \textcolor{w h i t e}{\text{dddd")->color(white)("dddd}} 4 x + 5 \left(\textcolor{red}{- \frac{4}{17}}\right) = - 4}$

$\textcolor{w h i t e}{\text{dddddddddddddddd")->color(white)("dddd")4xcolor(white)("ddd")-20/17color(white)("dd}} = - 4$

Add $\frac{20}{17}$ to both sides

$\textcolor{w h i t e}{\text{dddddddddddddddd")->color(white)("dddd")4xcolor(white)("dd}} = - 4 + \frac{20}{17}$

$\textcolor{w h i t e}{\text{dddddddddddddddd")->color(white)("dddd}} 4 x = - \frac{48}{17}$

Divide both sides by 4

$\textcolor{w h i t e}{\text{ddddddddddddddd")->color(white)("dddd}} x = - \frac{12}{17}$