https://file.aiquickdraw.com/mathbot/file/1727182417305rjgew4qk.jpg

The system of equations provided in the image is:

\[
\begin{aligned}
1. & \quad 12x - 3y + 6z = -18 \\
2. & \quad 3x + 3y - 4z = 44 \\
3. & \quad 15x + 2z = 26
\end{aligned}
\]

We can solve this system of three equations in three variables \(x\), \(y\), and \(z\) using substitution or elimination methods. Let me solve this step by step.

### Step 1: Solve for \(z\) from the third equation.
From equation (3):
\[
15x + 2z = 26 \implies z = \frac{26 - 15x}{2}
\]

### Step 2: Substitute \(z\) into equations (1) and (2).

#### Substituting \(z = \frac{26 - 15x}{2}\) into equation (1):
\[
12x - 3y + 6\left(\frac{26 - 15x}{2}\right) = -18
\]
Simplify:
\[
12x - 3y + 3(26 - 15x) = -18
\]
\[
12x - 3y + 78 - 45x = -18
\]
\[
-33x - 3y + 78 = -18
\]
\[
-33x - 3y = -96 \quad \text{(Divide by -3)}
\]
\[
11x + y = 32 \quad \text{(Equation A)}
\]

#### Substituting \(z = \frac{26 - 15x}{2}\) into equation (2):
\[
3x + 3y - 4\left(\frac{26 - 15x}{2}\right) = 44
\]
Simplify:
\[
3x + 3y - 2(26 - 15x) = 44
\]
\[
3x + 3y - 52 + 30x = 44
\]
\[
33x + 3y - 52 = 44
\]
\[
33x + 3y = 96 \quad \text{(Divide by 3)}
\]
\[
11x + y = 32 \quad \text{(Equation B)}
\]

### Step 3: Solve the system.
Both equations (A) and (B) are the same:
\[
11x + y = 32
\]
Thus, we have one equation left to solve for two variables. This implies that the system is dependent, meaning there are infinitely many solutions along a line in 3D space. We can express \(y\) in terms of \(x\) as follows:
\[
y = 32 - 11x
\]

### Step 4: Find \(z\) in terms of \(x\).
Now substitute this expression for \(x\) into the formula for \(z\):
\[
z = \frac{26 - 15x}{2}
\]

Thus, the solution can be written parametrically as:
\[
x = x, \quad y = 32 - 11x, \quad z = \frac{26 - 15x}{2}
\]

This describes the solution set for the system.

Would you like any further details on this solution, or do you have any questions?

Here are 5 related questions to think about:
1. How would the solution change if one of the equations was slightly altered?
2. How can the parametric solution be visualized in 3D space?
3. What method would you use to solve the system if a numerical solution was preferred?
4. Can this system of equations be solved using matrices or determinants?
5. What does the geometric interpretation of a dependent system of equations in three variables look like?

**Tip:** Always check for consistency when two or more equations result in the same relation, as this often indicates dependent systems or infinite solutions.

Solve the following system of equations:
12x - 3y + 6z = -18
3x + 3y - 4z = 44
15x + 2z = 26

Learn how to solve a system of three equations in three variables. The system provided has infinitely many solutions. The solution method includes substitution and simplification of linear equations. Suitable for high school algebra students.

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