# How do you solve the following system?: 6x -3y =-3 , -3x +y = 8

May 6, 2018

$x = - 7 , y = - 13$ fulfills the two equations.

#### Explanation:

You can do it two ways. One is graphical:
Let the first graph be $6 {x}_{1} - 3 {y}_{1} + 3 = 0$ (where I have added 3 to each side of the first equation)
This can be arranged as the linear graph
${y}_{1} = 2 {x}_{1} + 1$ (divide all the terms with 3)

The other is
$- 3 {x}_{2} + {y}_{2} - 8 = 0$ or ${y}_{2} = 3 {x}_{2} + 8$

Graphically this gives: The solution that fulfills both equations then is x=-7, y=-13.

We can also solve it this way:

$6 x - 3 y + 3 = 0$
This gives $y = 2 x + 1$

Insert this in the other equation:
$- 3 x + y - 8 = - 3 x + \left(- 2 x + 1\right) - 8 = x - 7 = 0$
Which gives $x = - 7$

Insert this in the first equation:

$y = 2 x + 1 = - 14 + 1 = - 13$

Therefore $x = - 7 , y = - 13$ is our answer.

May 6, 2018

#### Explanation:

Isolate one variable in one of the equations, then use the expression for that variable to solve for the other. Plug the value of the variable you just solve for into either of the other expressions to find out the value of the other. I'll demonstrate below.

Step 1: Find an expression for $y$

(I'd recommend using the blue equation to do this step)

$6 x - 3 y = - 3$
$\textcolor{b l u e}{- 3 x + y = 8}$

$- 3 x \textcolor{b l u e}{+ 3 x} + y = 8 \textcolor{b l u e}{+ 3 x}$

$y = 3 x + 8$

Step 2: use the expression for $y$ to solve for $x$

$6 x - 3 \left(3 x + 8\right) = - 3$
$6 x - 9 x - 24 = - 3$
$- 3 x - 24 = - 3$

I've condensed all the various algebraic steps to solve for $x$ into one equation:

$\frac{- 3 x}{-} 3 = \frac{21}{-} 3$

$x = - 7$

Step 3: Use the found value of $x$ to solve for $y$ (I recommend using the blue equation).

$y = 3 x + 8$

$y = - 21 + 8$

$y = - 13$

Solution:

$x = - 7 , y = - 13$

Hope that helps!