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Solve the system.Solve the system. 2x-2y+z=5 2x-2y+3z=7 -3x-3y+3z=9

To solve the system of equations provided in the images, we have the following equations:

1. \(2x - 2y + z = 5\)
2. \(2x - 2y + 3z = 7\)
3. \(-3x - 3y + 3z = 9\)

Let's solve this system step-by-step:

### Step 1: Subtract the first equation from the second

Subtract equation (1) from equation (2):

\[
(2x - 2y + 3z) - (2x - 2y + z) = 7 - 5
\]

Simplify:

\[
2z = 2 \implies z = 1
\]

### Step 2: Substitute \(z = 1\) into the first and third equations

Substitute \(z = 1\) into equation (1):

\[
2x - 2y + 1 = 5 \implies 2x - 2y = 4
\]

Divide by 2:

\[
x - y = 2 \implies x = y + 2
\]

Substitute \(z = 1\) into equation (3):

\[
-3x - 3y + 3(1) = 9 \implies -3x - 3y + 3 = 9
\]

Simplify:

\[
-3x - 3y = 6
\]

Divide by -3:

\[
x + y = -2
\]

### Step 3: Solve for \(x\) and \(y\)

Now, we have two equations:

1. \(x - y = 2\)
2. \(x + y = -2\)

Add these two equations:

\[
(x - y) + (x + y) = 2 + (-2)
\]

\[
2x = 0 \implies x = 0
\]

Substitute \(x = 0\) into \(x + y = -2\):

\[
0 + y = -2 \implies y = -2
\]

### Step 4: Solution

Now we have all values:

\[
x = 0, \quad y = -2, \quad z = 1
\]

Thus, the solution to the system of equations is:

\[
(x, y, z) = (0, -2, 1)
\]

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. How would you verify the solution for a system of linear equations?
2. Can you solve the system of equations using a different method, such as matrix operations?
3. How does substitution work when solving systems of equations?
4. What is the significance of having a unique solution in a system of linear equations?
5. How can we determine if a system of equations has no solution or infinitely many solutions?

**Tip:** When solving systems of equations, always check if the solution satisfies all original equations. This helps confirm the accuracy of your solution.

Learn how to solve the system of equations 2x - 2y + z = 5, 2x - 2y + 3z = 7, -3x - 3y + 3z = 9 with a detailed step-by-step solution. This problem covers systems of linear equations and matrix operations. Suitable for students in Grades 11-12.

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