https://file.aiquickdraw.com/mathbot/file/172706283296143dcnfqc.jpg

The system of linear equations (SPLTV) shown in the image is:

1. \( x + 2z = 3y + 2 \)
2. \( y - z = -4x - 7 \)
3. \( 3z - 2 = -2(x + y - 10) \)

We need to solve this system of equations to find the values of \(x\), \(y\), and \(z\).

### Step 1: Simplify the equations
Let's start by simplifying each equation.

#### Equation 1:
\[ x + 2z = 3y + 2 \]
This is already simplified.

#### Equation 2:
\[ y - z = -4x - 7 \]
This is also in a simplified form.

#### Equation 3:
\[ 3z - 2 = -2(x + y - 10) \]
Expand the right-hand side:
\[
3z - 2 = -2x - 2y + 20
\]
Now move all terms to one side:
\[
3z + 2x + 2y = 22
\]
This is the simplified form of the third equation.

### Step 2: Solve the system
We now have the simplified system:
1. \( x + 2z = 3y + 2 \)
2. \( y - z = -4x - 7 \)
3. \( 3z + 2x + 2y = 22 \)

Next, I will solve this system of equations step-by-step.

Let me proceed with the calculations to solve for \(x\), \(y\), and \(z\).The solution to the system of equations is:

\[
x = -1, \quad y = 3, \quad z = 6
\]

Would you like further details or clarifications on any of the steps? Here are some related questions to consider:

1. How can I verify the solution by substituting it back into the original equations?
2. What is the general method to solve systems of linear equations?
3. Can this system be solved using matrix methods, such as Gaussian elimination?
4. How does substitution work in solving systems like this one?
5. What are other types of systems of equations, and how are they solved?

**Tip:** For systems of linear equations, it's often helpful to rearrange each equation in terms of one variable to simplify substitution and reduce complexity.

Solve the system of linear equations (SPLTV) consisting of the following: 1. x + 2z = 3y + 2, 2. y - z = -4x - 7, 3. 3z - 2 = -2(x + y - 10).

Learn how to solve a system of linear equations with three variables (SPLTV). This problem involves simplifying and solving three equations step-by-step, suitable for students in Grades 10-12.

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