# How do you solve the following system?: -5x -2y =3 , 2x -y = -6

Nov 12, 2015

#### Answer:

Shown two methods. The first in detail. Solved for $x$ and left finding the value of $y$ for the question proposer.

#### Explanation:

$\textcolor{b r o w n}{\text{You have 2 unknowns and 2 equations so it is solvable.}}$

$\textcolor{b r o w n}{\text{Two ways of solving:}}$

## $\textcolor{b l u e}{\text{Method 1}}$

In detail: every step shown!!
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Consider $- 5 x - 2 y = 3$ making $y$ the dependant variable

$\textcolor{b r o w n}{\text{Showing the method for changing the variable for "5x-2y=3" only.}}$

Add $\textcolor{g r e e n}{2 y}$ to both sides giving:
$\left(- 5 x - 2 y\right) \textcolor{g r e e n}{+ 2 y} = \left(3\right) \textcolor{g r e e n}{+ 2 y}$

$- 5 x = 3 + 2 y$

Subtract $\textcolor{g r e e n}{3}$ from both sides giving:
$\left(- 5 x\right) \textcolor{g r e e n}{- 3} = \left(2 y + 3\right) \textcolor{g r e e n}{- 3}$

$- 5 x - 3 = 2 y$

Divide both sides by $\textcolor{g r e e n}{2}$
Note that divide by 2 is the same as multiply by $\frac{1}{2}$

$\textcolor{g r e e n}{\frac{1}{2}} \left(- 5 x - 3\right) = \textcolor{g r e e n}{\frac{1}{2}} \left(2 y\right)$

$- \frac{5}{2} x - \frac{3}{2} = y$
Write as $\textcolor{b l u e}{y = - \frac{5}{2} x - \frac{3}{2} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \left(1\right)}$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b r o w n}{\text{Changing the variable for } 2 x - y = - 6}$

$\textcolor{b l u e}{y = 2 x + 6. \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \left(2\right)}$

Subtract $\textcolor{g r e e n}{\left(1\right)}$ from (2) giving:

$y - \textcolor{g r e e n}{y} = 2 x - \textcolor{g r e e n}{\left(- \frac{5}{2} x\right)} + 6 - \textcolor{g r e e n}{\left(- \frac{3}{2}\right)}$
$0 = \frac{9}{2} x + \frac{15}{2}$
$\frac{9}{2} x = - \frac{15}{2}$

Multiply throughout by 2 giving:
$9 x = 15$

$\textcolor{b l u e}{x = \frac{15}{9.} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . . \left(3\right)}$

Next step I will leave for you to do!
Substitute (3) into either of (1) or (2) to find the value of y

## $\textcolor{b l u e}{\text{Method 2}}$

$- \frac{5}{2} x - \frac{3}{2} = y = 2 x + 6$

$- \frac{5}{2} x - \frac{3}{2} = 2 x + 6$

Now solve for x and substitute in (1) or (2) to find y

$\textcolor{red}{\text{Both methods are fast once you are}}$

$\textcolor{red}{\text{able to "underline("change the variable by sight}}$