How do solve the following linear system?:  -2x+3y=-1 , -x-7y=14 ?

Aug 29, 2016

First, solve equation #2 for x.

Explanation:

This gives you $x = - 7 y - 14$ .

Now substitute this value of x into the 1st equation, like this:

$- 2 \left(- 7 y - 14\right) + 3 y = - 1$

Multiply the negative 2 into the parentheses:

$14 y + 28 + 3 y = - 1$

Combine like terms:

$17 y + 28 = - 1$

Move the 28 to the other side of the equation:

$17 y = - 29$

Divide both sides by 17 to isolate the y-term:

$17 \frac{y}{17} = - \frac{29}{17}$

which gives you:

$y = - \frac{29}{17}$.

Since 17 and 29 are both prime numbers, the answer cannot be reduced.

Now you have the value of y. Plug that into the either eqn:

$- 2 x + 3 \left(- \frac{29}{17}\right) = - 1$

and solve for x.

$- 2 x - \frac{87}{17} = - 1$
$- 2 x = - 1 + \frac{87}{17} = \frac{70}{17}$

Divide both sides by negative 2:
$x = \frac{\frac{70}{17}}{-} 2 = \frac{70}{17} \cdot \left(- \frac{1}{2}\right) = - \frac{70}{34}$

Reduce:
$x = - \frac{35}{17}$

Now plug $x = - \frac{35}{17}$ and $y = - \frac{29}{17}$ into either eqn to check the answers:

Does $- 2 \left(- \frac{35}{17}\right) + 3 \left(- \frac{29}{17}\right) = - 1$ ?
$\frac{70}{17} - \frac{87}{17} = - 1$ ?
$- \frac{17}{17} = - 1$

The fractions look odd, but it checks out.