# How do you solve 4x + y = -9 and 3x + 7y = 37?

Jun 12, 2016

Solve one equation for x in terms of y, then substitute in the result for x in the second equation. Solve second equation for y, then plug in the value for y into either equation to solve for x.
x = -4 y = 7

#### Explanation:

Equations:
1) $4 x + y = - 9$
2) $3 x + 7 y = 37$

If we wished to do so here, we could multiply equation 2 by 4/3, and subtract the result from equation 1 (or subtract equation 1 from the result). However, that will require substantial dealing with fractions. The below method should help us avoid some of that.

a) Solve either equation for one variable.
In this case, the easiest way to do this is to solve (1) for $y$.

$4 x + y = - 9 \implies y = - 4 x - 9$

b) Plug the solution for y into the other equation.

Since from (a) we have $y = - 4 x - 9$...
$3 x + 7 y = 37$
$3 x + 7 \left(- 4 x - 9\right) = 37$
$3 x - 28 x - 63 = 37$
$- 25 x = 100$
$x = - 4$

c) Now that we have a solution for x, we can plug it into either of our initial equations (or into our revised version of equation 1, which put y in terms of x) to obtain a value for y. Below we use the revised equation 1 for simplicity's sake.

$y = - 4 x - 9$
$y = - 4 \left(- 4\right) - 9$
$y = 16 - 9$
$y = 7$

Thus our solution is $\left(x , y\right) = \left(- 4 , 7\right)$