# How do you solve the system  -7x+y=-19 and -2x+3y=-19?

Apr 7, 2018

$\left(2 , - 5\right)$

Graphically: #### Explanation:

There's two ways in which we solve systems in general: elimination, and substitution.

We'll be using substitution to solve this system. Why? Notice that we have a single $y$ term in the first equation, which makes for a relatively straightforward substitution. So, let's walk through this:
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Step 1: Solve for One Variable
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Let's first write out our equations:

(1) $- 7 x + y = - 19$
(2) $- 2 x + 3 y = - 19$

Now, we solve for one variable. I'm going to solve for $y$ in equation (1):

$\implies - 7 x + y = - 19$
$\implies \textcolor{red}{y = 7 x - 19}$

As you can see, that was pretty easy, and gave us a relatively nice result. This is why we chose to do substitution for this particular problem.
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Step 2: Plug into Other Equation; Solve for Other Variable.
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Now, let's plug in the value for $y$ we procured above into equation (2):

$\implies - 2 x + 3 \textcolor{red}{\left(7 x - 19\right)} = - 19$

Foil:
$\implies - 2 x + 21 x - 57 = - 19$

Note: Watch your signs while you do this

Combine like terms:
$\implies 19 x - 57 = - 19$

Isolate $x$:
$\implies 19 x = 38$
$\implies x = \frac{38}{19} = \textcolor{b l u e}{2}$
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Step 3: Solve for First Variable
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We could plug this value we found for $x$ into either of our initial equations, and solve for $y$. However, we can save ourselves some extra algebra by plugging it into our substitution for $y$, found in step 1:

$y = 7 x - 19$

$\implies y = 7 \textcolor{b l u e}{\left(2\right)} - 19$
$\implies y = 14 - 19 = \textcolor{red}{- 5}$

So, our final solutions are $\textcolor{b l u e}{x = 2}$ and $\textcolor{red}{y = - 5}$. In other words, the solution to this equation is represented by the point $\left(2 , - 5\right)$

You can see this graphically below. The red line is equation (1) and the blue line is equation (2): Hope that helped :)