Question

What is the value of y in the system of equations below?
3x+y+2z=−10
x+5y−2z=−12
−2x+3y=−1

Answer 1

To find the value of y, we need to solve the system of equations. Here is one way to do it:

First, let's solve the third equation for x:
-2x + 3y = -1
-2x = -1 - 3y
x = (1 + 3y) / -2

Now, substitute this value of x into the second equation:
(1 + 3y) / -2 + 5y - 2z = -12
Multiply through by -2 to get rid of the fraction:
1 + 3y - 10y + 4z = 24
Combine like terms:
4z - 7y = 23 ----(1)

Next, substitute the expression for x into the first equation:
3 * ((1 + 3y) / -2) + y + 2z = -10
Multiply through by -2 to get rid of the fraction:
-6 - 18y + -2y + -4z = 20
Combine like terms:
-20z - 20y = 26 ----(2)

Now we have a system of two equations:
4z - 7y = 23 ----(1)
-20z - 20y = 26 ----(2)

To solve this system, we can multiply the first equation by -5 and add it to the second equation:

-5(4z - 7y) + (-20z - 20y) = -5(23) + 26
-20z + 35y -20z - 20y = -115 + 26
-40z + 15y = -89

Divide through by 5 to get rid of the coefficient of y:
-8z + 3y = -17 ----(3)

Now we have a system of two equations:
-8z + 3y = -17 ----(3)
-20z - 20y = 26 ----(2)

Multiply equation (3) by 7 and add it to equation (2):

7(-8z + 3y) + (-20z - 20y) = 7(-17) + 26
-56z + 21y -20z - 20y = -119 + 26
-76z + y = -93

Since we are trying to find the value of y, we can isolate it by adding 76z to both sides:
y = -93 + 76z

So, the value of y in the system of equations is -93 + 76z. The specific value of y would depend on the value of z.