Question

Note: Enter your answer and show all the steps that you use to solve this problem in the space provided. What is the solution to the following system of equations? x - y +z -1 x+y+3z=-3 2x-y+2z=0

Answer 1

To solve the system of equations:

x - y + z - 1 = 0 ---(1)
x + y + 3z = -3 ---(2)
2x - y + 2z = 0 ---(3)

We can solve this system of equations using elimination or substitution method. Let's use the elimination method.

First, we can eliminate the variable y from equations (1) and (3) by adding the two equations together:

(x - y + z - 1) + (2x - y + 2z) = 0 + 0
3x + 3z - 2y - 1 = 0 ---(4)

Now, let's eliminate the variable y from equations (2) and (4) by multiplying equation (2) by 2:

2(x + y + 3z) = 2(-3)
2x + 2y + 6z = -6 ---(5)

Next, we can eliminate the variable x from equations (4) and (5) by subtracting the two equations:

(3x + 3z - 2y - 1) - (2x + 2y + 6z) = 0 - (-6)
3x - 2x + 3z - 2y - 6z + 2y - 1 + 6 = 0 + 6
x - 3z + 7 = 0 ---(6)

Now, we have two equations with two variables: equations (6) and (2):

x - 3z + 7 = 0 ---(6)
x + y + 3z = -3 ---(2)

We can solve equation (6) for x:

x = 3z - 7

Substituting this value of x into equation (2), we get:

(3z - 7) + y + 3z = -3
6z + y - 7 = -3
6z + y = 4 ---(7)

Now, we have equations (7) and (6):

6z + y = 4 ---(7)
x = 3z - 7 ---(6)

We have two equations with two variables. We can solve this system of equations by substitution.

Substituting x = 3z - 7 into equation (1), we get:

(3z - 7) - y + z - 1 = 0
4z - y - 8 = 0
4z - y = 8 ---(8)

Now, we have equations (8) and (7):

4z - y = 8 ---(8)
6z + y = 4 ---(7)

Adding equation (8) and (7) together, we eliminate y:

(4z - y) + (6z + y) = 8 + 4
10z = 12

Dividing both sides by 10, we can solve for z:

z = 12/10
z = 6/5

Substituting this value of z back into equation (7), we can solve for y:

6(6/5) + y = 4
36/5 + y = 4
y = 4 - 36/5
y = (20 - 36)/5
y = -16/5

Substituting the values of y and z into equation (6), we can solve for x:

x = 3(6/5) - 7
x = 18/5 - 35/5
x = (18 - 35)/5
x = -17/5

Therefore, the solution to the system of equations is:
x = -17/5
y = -16/5
z = 6/5