Armani, I will not do every problem you have, but, if you carefully review my answer to this question and read all my words, you should be able to do your own work on the other problems, which are similar. It's not about the final answer; it about UNDERSTANDING THE PROCESS.
First, understand that you were given two equations, each of which define a line in the xy-plane. These lines are identical or parallel (meaning they don't intersect), or they intersect at a point. When we are asked to solve a system of equations, we are asked to figure out that point at which the lines intersect. Or, if there is NO solution, then the lines are parallel. If there are an infinite number of points, then the lines are identical.
You can solve systems of equations by graphing, by elimination, or by substitution. Often, the strategy is up to you, but, in this case, you have been asked to solve by elimination.
Here are your two linear equations:
4x − 3y = 8
5x − 2y = −11
When we use the elimination technique, we get rid of one of the variables (x or y) by adding or subtracting a version of one equation to a version of the other. If we were to use the original equations as written, neither x nor y would drop out. Thus, we need to multiply the equations by a number that will cause x or y to drop out. There are multiple ways to do this even within this problem. However, I am going to multiply the first equation by (–5) and the second equation by 4 and then add the new equations (same lines) and the x terms will drop out.
(–5)(4x − 3y) = 8(–5) ==>> –20x + 15y = –40
4(5x − 2y) = −11(4) ==>> 20x – 8y = –44
Now to add these new equations to get:
7y = –84
Divide by 7 to see what 1y is:
7y = –84
___ ___
7 7
y = –12
We got y by ELIMINATING x.
Now we are going to get x by SUBSTITUTING the value of y in one of the original equations. I'm going to pick the first equation, but it doesn't matter which you choose.
4x − 3y = 8
y = –12
4x – 3(–12) = 8
4x – (–36) = 8
4x + 36 = 8
4x + 36 – 36 = 8 – 36
4x = –28 (Now divide each side by 4)
x = –7
The intersection point of the two lines—or, you can say, the solution of the system of linear equations—is
(–7,–12).
