SOLUTION: y=-6x+4 y=2x-12

Question 686094: y=-6x+4
y=2x-12
Answer by RedemptiveMath(80) (Show Source): You can put this solution on YOUR website!
All you need to do is set both equations equal to each other. This can be done by replacing one of the y's with the other equation. These two equations express a similar fact: The first equation says that y equals a variable x multiplied by -6 while being added by 4, but the second equation says that y also equals the same variable x multiplied by 2 and subtracted by 12. Since this information says that y equals two things, we can just set those two things, or equations, equal to each other. (Y can only equal one number; the expressions "-6x+4" and "2x-12" must equal that number since y equals both of these expressions.)

So, setting the expressions equal to each other, we solve for the remaining variable x:

-6x+4 = 2x-12
+6x +6x (add 6x to each side of the equation)
4 = 8x-12
+12 +12 (add 12 to each side of the equation)
16 = 8x
- -
8 8 (divide each side of the equation by 8)
2 = x or x = 2.

We first can add 6x to each side of the equation in order to get x and all of its components to one side of the equation. The goal is to eventually get x by itself with nothing else next to it, but this is a first step. When we add 6x to each side of the equation, we "cancel" the 6x on the left side but make 2x turn into 8x on the right. Whenever we want to cancel a term when trying to solve for a variable, we use the opposite operation of that term on both sides of the equation. Since 6x is negative initially, we use a positive 6x or we add 6x to both sides. This technically leaves 0x on the left side, but anything multiplied by 0 is 0. You can leave 0 when moving the equation around, but it really makes no difference when the remaining operations are addition or subtraction as in this case. Next, we add 12 to each side of the equation to get rid of -12 and allow the variable and its components on one side. This gives 16 on the left side. Finally, we just have to get x by itself. Since 8x means "8 times x," we have to do the opposite operation of division. Dividing both sides of the equation of 8 will give us the answer of x = 2.

We have x = 2, but this is not the entire answer. If you're solving for system of equations, you need to generally find what both variables equal. All we have to do is plug what we have for the first variable x into the original equations. We don't have to do it for both for either one will work.

y = -6x+4
y = -6(2)+4
y = -12 + 4
y = -8
OR
y = 2x-12
y = 2(2)-12
y = 4-12
y = -8.

As you can see, both equations will give y = -8 when 2 is substituted for x. However, this still is usually not the entire answer. When working with two variables (generally x and y) with no other information from a story, the problem usually wants the answer written as a point. Points are written as sets of numbers using the Cartesian coordinate system. Writing x first and y second, we have the point (2, -8). If the problem does not want the answer written as a point, then x = 2 and y = -8 will work.

To explain how we can manipulate equations like I did below the second paragraph, you must understand the concept of equality. Equality allows us to manipulate equations as long as every component of the equations are manipulated exactly the same way. If this is done, then the equations technically remain the same in the sense of input/output. Although the equations will look differently, whatever x and y equals will not change. X is what we consider an input and y is what we consider an output in the Cartesian coordinate system.
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