Question

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

Answer 1

`(2, -5)`

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:


Step 1: Solve for One Variable
--

Let's first write out our equations:

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

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

`=> -7x + y = -19`
`=> color(red)(y = 7x - 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.


Step 2: Plug into Other Equation; Solve for Other Variable.
--
Now, let's plug in the value for `y` we procured above into equation (2):

`=> -2x + 3color(red)((7x - 19)) = -19`

Foil:
`=> -2x + 21x - 57 = -19`

Note: Watch your signs while you do this

Combine like terms:
`=> 19x - 57 = -19`

Isolate `x`:
`=> 19x = 38`
`=> x = 38/19 = color(blue)(2)`


Step 3: Solve for First Variable
--
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 = 7x - 19`

`=> y = 7color(blue)((2)) - 19`
`=> y = 14 - 19 = color(red)(-5)`

So, our final solutions are `color(blue)(x = 2)` and `color(red)(y = -5)`. In other words, the solution to this equation is represented by the point `(2,-5)`

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

Hope that helped :)