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

1 Answer
Apr 7, 2018

Answer:

#(2, -5)#

Graphically:

Created by Darshan Senthil (on desmos)

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) #-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.
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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):

#=> -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)#
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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 = 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):

Created by Darshan Senthil (on desmos)

Hope that helped :)