I need help to figure this out.

Question

8x+6y=-40

-2x-4y=10

Answer 1

If you have an equation, you can multiply both sides by anything (as long as it's the same thing) and get an equation that is still true.

 

If you have an equation, you can add anything (as long as it's the same thing) to both sides and get an equation that is still true.

 

This is why we can solve a system of equations like this by elimination. 

 

Here's an example:

 

x + y = 1

x - y = 3

 

If I add these equations vertically, this is a valid thing to do because you can think of it as adding the same thing to both sides of the top equation. 

 

If I could add 3 to both sides of the top equation, then I can also add 3 to the right side and x - y to the left side, and even though x - y looks like it's a different thing, it isn't because we have from the second equation that x - y is the same thing as 3.

 

So add the left hand sides together and add the right hand sides together to get:

 

2x = 4

 

Then x has to be 2 which means, according to the first equation, x + y = 1, that 2 + y = 1. According to the second equation, x - y = 3, this means that 2 - y = 3. Solve either of these equations for y to get that y has to be -1.

 

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So, to solve these:

 

1) Find some clever way to get a variable by itself and solve for it, and

2) Plug this into any equation you have containing both of them to get the other

 

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8x + 6y = -40

-2x - 4y = 10

 

Note here that if I multiply the second equation by 4, I will get the x terms to be opposites of each other like the ys were in the example:

 

8x + 6y = -40

 

-8x - 16y = 40

 

Add the left sides together and the right sides together to get

 

-10y = 0

 

The only thing I can multiply -10 by and get 0 is 0. So y must be 0.

 

y = 0

 

Pick one of the equations above and rewrite it letting y be 0:

 

8x + 6(0) = -40

 

8x = -40

 

I have to multiply 8 by -5 to get -40, so x has to be -5.

 

x = -5

 

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We have just found a point that makes both of those statements true. 

 

Remember that these are linear equations. If you were to graph them, the point (-5, 0) would be where those lines cross.

 

 

   

Answer 2

To solve by substitution, you don't necessarily have to get x or y by itself.  It can be some multiple of either.

For example, I can multiply the second equation by (-4) to get:

8x+16y = -40, and then subtract the 16y to the other side:

8x = -40-16y.  Now I substitute out what '8x' is equal to in the first equation:

(-40-16y)+6y = -40.  In this case, I can just drop the parentheses:

-40-16y+6y = -40.  Add 40 to both sides, and combine the y-terms:

-10y = 0. Divide both sides by -10 to get:

y=0.

 

Plug this value for y back into either of the original equations.  I am going to use the 2nd:

-2x-4(0) = 10, equivalent to:

-2x = 10.  Divide both sides by -2:

x = -5

 

Answer: (-5,0)

Answer 3

-2x - 4y = 10                divide both sides by -2

x + 2y = -5                   subtract 2y from both sides

x = -2y - 5                    substitute for x in other equation

8(-2y - 5) + 6y = -40    distribute

-16y - 40 + 6y = -40    combine like terms

-10y - 40 = -40             add 40 to both sides

-10y = 0                       divide both sides by -10

y = 0                            substitute in either equation 

8x + 6(0) = -40

8x = -40

x = -5                           or

 

-2x - 4(0) = 10

-2x = 10

x = -5

 

{-5, 0}