How do you solve 3x+2y = -9 and -10x + 5y = - 5?

Jul 19, 2017

There are two different methods: substitution and elimination. The result is $x = - 1$ and $y = - 3$

Explanation:

You have to solve them as simultaneous equations. There are two methods that can work: substitution and elimination. Personally, however, I prefer elimination (and I'd prefer not to explain two concepts in a single explanation) so that is what I will explain below.

(I may choose to add an explanation about the substitution method separately)

First, however, we can simplify the second equation...
$- 10 x + 5 y = - 5$ divided by 5 on both sides, becomes
$- 2 x + y = - 1$

Then..

In the elimination method we will try to eliminate one variable by subtracting it from both equations. After solving for the variable that's left, we substitute back into the given equations to solve for the eliminated variable.

Our equations are:
1. $3 x + 2 y = - 9$
2. $- 2 x + y = - 1$

We want to see how we can make one of these variable have equal coefficients in both equations. Looking above, you may notice that this is easy to do with variable y, by multiplying equation 2, by 2...

1. $3 x + 2 y = - 9$
2. $- 4 x + 2 y = - 2$

Now we have +2y in both equations. Now we'll want to subtract equation 1 - equation 2. Or, in order to avoid confusion, you can multiply the bottom equation by -1, then add the equations together.

$3 x + 2 y = - 9$
+
$4 x - 2 y = 2$
gives
$7 x = - 7$
$x = - 1$

Now that we have the value for x we can substitute it into one of our original equations to find y.

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

And so, we have our solution of $x = - 1$ and $y = - 3$