# How do you solve the following linear system:  -x+2y=-6 , 5x-2y=-35 ?

Apr 26, 2018

Isolate $x$ , replace $x$ with the expression equal to $x$ , isolate $y$, and then use the value of $y$ to solve for $x$.

#### Explanation:

Isolate $x$ in the simplest manner possible:

Equation 1: $- x + 2 y = - 6$
Equation 2: $5 x - 2 y = - 35$

1. $- x + x + 2 y = - 6 + x$
2. $2 y + 6 = - 6 + x + 6$
3. $2 y + 6 = x$

Substitute the expression in step 3 in for $x$ in equation 2

$2 y + 6 = x \to 5 \left[2 y + 6\right] - 2 y = - 35$

Distribute $5$ onto $2 y + 6$ to get the equation,

$10 y + 30 - 2 y = - 35$

Which becomes...

$8 y = - 65$

Now isolate $y$ ...

$\frac{8 y}{8} = - \frac{65}{8}$

$y = - \frac{65}{8}$

... and use the value of $y$ to find $x$ !

$- x - 2 \left(\frac{- 65}{8}\right) = - 6$

$- 2 \cdot - \frac{65}{8} = \frac{130}{8} = 16.25$
$- x = - 6 + \frac{130}{8} \to - x = \frac{82}{8}$

There aren't negative signs on both sides of the equation, so the value of $x$ is in turn, positive, and the right side of the equation, becomes negative:

$x = - \frac{82}{8} , y = - \frac{65}{8}$

Since these are linear systems, these are the only solutions to the system. You can always rewrite both equations in terms of $y$ and graph them. The x and y coordinates where the two lines intersect one another $\left(- \frac{82}{8} , - \frac{65}{8}\right)$, are the solutions to the system.

Graph these:
$y = \frac{5}{2} x + \frac{35}{2}$
$y = \frac{1}{2} x - 3$