# How do you solve the following linear system:  -x+3y=-9 , 8x-4y=32 ?

Apr 10, 2018

$y = - 2$
$x = 3$

#### Explanation:

We have, $3 y - x + 9 = 0$ and $4 \left(2 x - y\right) = 4 \times 8$
Or, $3 \left(y + 3\right) = x$ and $2 x - y = 8$
Substituting the value of $x$ from Equation 1 in Equation 2,we get,
$2 \left(3 y + 9\right) - y - 8 = 0$
Or, $6 y - y + 18 - 8 = 0$
Or, $5 y = - 10$
Or, $y = - 2$
Thus, $x$ comes out to be $3 \times \left(3 - 2\right)$ or $3 \times 1$ or $3$

Apr 10, 2018

$x = 3$

$y = - 2$

#### Explanation:

Given:

$- x + 3 y = - 9 \text{ } \ldots \ldots \ldots \ldots \ldots \ldots \ldots . E q u a t i o n \left(1\right)$
color(white)("d")8x-4y=32" ".........................Equation(2)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine the value of "y" - in detail using first principles}}$

$\textcolor{b r o w n}{\text{Consider } E q n \left(1\right)}$

$\textcolor{b r o w n}{\text{Multiply both sides by } \left(- 1\right)}$

$+ x - 3 y = + 9$

$\textcolor{b r o w n}{\text{Add "3y" to both sides}}$

$x = 9 + 3 y \text{ } \ldots \ldots . E q u a t i o n \left({1}_{a}\right)$

$\textcolor{b r o w n}{\text{Using "Eqn(1_a)" substitute for "x" in } E q n \left(2\right)}$

$\textcolor{g r e e n}{8 \textcolor{red}{x} - 4 y = 32 \textcolor{w h i t e}{\text{dddd")->color(white)("dddd}} 8 \left(\textcolor{red}{9 + 3 y}\right) - 4 y = 32}$

$\textcolor{g r e e n}{\textcolor{w h i t e}{\text{dddddddddddddd.")->color(white)("dddd}} 72 + 24 y - 4 y = 32}$

$\textcolor{g r e e n}{\textcolor{w h i t e}{\text{dddddddddddddd.")->color(white)("dddd")72+20ycolor(white)("dddd}} = 32}$

$\textcolor{b r o w n}{\text{Subtract 72 from both sides}}$

$\textcolor{g r e e n}{\textcolor{w h i t e}{\text{dddddddddddddd.")->color(white)("dddddddd")20ycolor(white)("dddd}} = - 40}$

$\textcolor{b r o w n}{\text{Divide both sides by 20}}$

$\textcolor{g r e e n}{\textcolor{w h i t e}{\text{dddddddddddddddd.")->color(white)("dddddddd")ycolor(white)("dddd}} = - 2}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine the value of "x" - not so much detail}}$

$\textcolor{b r o w n}{\text{Substitute for "y" in } E q n \left(1\right)}$

$- x + 3 y = - 9 \textcolor{w h i t e}{\text{dddd")->color(white)("dddd}} - x + 3 \left(- 2\right) = - 9$

$\textcolor{w h i t e}{\text{ddddddddddddddddddddddddd")->color(white)("dddd}} - x = - 3$

$x = 3$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Check}}$

$- x + 3 y \textcolor{w h i t e}{\text{d")->color(white)("d}} - 3 + \left(3 \times - 2\right) \to 9$ as required

$8 x - 4 y \textcolor{w h i t e}{\text{d")->color(white)("d}} 8 \left(3\right) - 4 \left(- 2\right) \to 24 + 8 \to 32$ as req'd