8. Solve the system using elimination.

x + 2y = 1
4x + y = 11

So I added both equations and got 5x + 3y = 12 but am unsure of where to go next with it.
I am having the same problem on another question just a bit different than this in which it just asks to solve for x.

you haven't eliminated anything. Adding the equations does no good unless x or y disappears. Try writing them as

x + 2y = 1
8x + 2y = 22

Now if you subtract them, y is eliminated...

To solve the system of equations using elimination, you want to eliminate one variable by adding or subtracting the equations. In this case, it looks like the coefficients of y in both equations are already the same, so we can use these two equations:

x + 2y = 1
4x + y = 11

First, let's multiply the second equation by 2 to make the coefficients of y the same:

2(4x + y) = 2(11)
8x + 2y = 22

Now, we have two equations:

x + 2y = 1
8x + 2y = 22

To eliminate the variable y, subtract the first equation from the second equation:

(8x + 2y) - (x + 2y) = 22 - 1
7x + 0y = 21
7x = 21

Now, we have a single equation with one variable:

7x = 21

To solve for x, divide both sides of the equation by 7:

7x/7 = 21/7
x = 3

Therefore, the value of x is 3.

To find the corresponding value of y, substitute the value of x back into one of the original equations. Let's use the first equation:

x + 2y = 1
3 + 2y = 1

Subtract 3 from both sides:

2y = 1 - 3
2y = -2

Divide both sides by 2:

y = -2/2
y = -1

Therefore, the solution to the system of equations is x = 3 and y = -1.

Now, let's consider the question where you only need to solve for x. The same principles apply. If you have a system with two equations, you can eliminate one variable and solve for the remaining variable. In this case, you will just have one equation with one variable left, and you can solve for x directly.