define the following new variables

X to A =W
X to B =X
Y to A =Y
Y to B =Z

w+x=32
y+z=8
w+y=22
x+z=18

You helped me solve for y Ijust added w+y=32

220w+300(32-w)+400(22-w)+180(8-y)=9280

,

w+y=22

The answer I got was y=32 I solved for x and got x=42 I don't know if this is right. I am trying to solve for w and z but keep coming up with fractions. I am using the substitution method to calculate. Thank you Can you help me understand.

It is not clear to me what this parts means:

"define the following new variables
X to A =W
X to B =X
Y to A =Y
Y to B =Z "

However, if we look at the system of 4x4 equations:
w+x=32
y+z=8
w+y=22
x+z=18

Whichever method you use, you will find that one of the equations is a linear combination of the other three, and there is an infinite number of solutions, depending on the value of the "free variable".

If we take the "free variable" as z, then we can express the solution in terms of z, as follows:
w=t+14, x=18-t, y=8-t, z=t

To solve for the variables w, x, y, and z in the given system of equations, we can use the substitution method. Let's go through the steps together:

1. We are given the equations:
w + x = 32 (Equation 1)
y + z = 8 (Equation 2)
w + y = 22 (Equation 3)
x + z = 18 (Equation 4)

2. You mentioned that you solved for y and got y = 32 by adding w + y = 32. However, this equation contradicts equation 3 (w + y = 22). Since these two equations cannot be true simultaneously, there seems to be an error in one of them. Let's stick to equation 3 for now: w + y = 22.

3. To solve for w, we can use equation 3 and subtract y from both sides:
w = 22 - y (Equation 5)

4. Now, let's substitute equation 5 into equation 1 to solve for x:
(22 - y) + x = 32
x = 32 - 22 + y
x = 10 + y (Equation 6)

5. Next, we can substitute equations 5 and 6 into equation 2 to solve for z:
y + z = 8
y + z = 8
y + (8 - y) = 8
y - y + z = 8 - 8
z = 0 (Equation 7)

6. Finally, we have found the values of w, x, y, and z:
w = 22 - y
x = 10 + y
y = y
z = 0

Please note that the equation you mentioned, "w + y = 32," is not consistent with the given system of equations, and it seems that there may have been a mistake in solving for y.

If you have any more specific questions or need further assistance, feel free to ask!