https://file.aiquickdraw.com/mathbot/file/1727308599811e39hi8u8.jpg

The system of equations provided in the image is:

\[
\begin{aligned}
-3x - 7y + 2z &= -5 \\
-3x + 8y - 10z &= 7 \\
3x + y + 4z &= -7 \\
\end{aligned}
\]

Let's solve this system step by step using methods like elimination or substitution.

### Step 1: Eliminate one variable

We can try to eliminate \(x\) first. Add the first and third equations:

\[
(-3x - 7y + 2z) + (3x + y + 4z) = -5 + (-7)
\]
This simplifies to:
\[
-6y + 6z = -12 \quad \text{or} \quad -y + z = -2 \quad \text{(Equation A)}
\]

Next, subtract the first equation from the second to eliminate \(x\):

\[
(-3x + 8y - 10z) - (-3x - 7y + 2z) = 7 - (-5)
\]
Simplifying this:
\[
8y + 7y - 10z - 2z = 12
\]
\[
15y - 12z = 12 \quad \text{(Equation B)}
\]

### Step 2: Solve the two-variable system

Now we have two equations:
1. \(-y + z = -2\) (Equation A)
2. \(15y - 12z = 12\) (Equation B)

We can solve these equations for \(y\) and \(z\). First, solve Equation A for \(z\):
\[
z = y - 2
\]

Substitute this into Equation B:
\[
15y - 12(y - 2) = 12
\]
Simplifying:
\[
15y - 12y + 24 = 12
\]
\[
3y + 24 = 12
\]
\[
3y = -12
\]
\[
y = -4
\]

### Step 3: Find \(z\)

Substitute \(y = -4\) into \(z = y - 2\):
\[
z = -4 - 2 = -6
\]

### Step 4: Find \(x\)

Now substitute \(y = -4\) and \(z = -6\) into the first equation:
\[
-3x - 7(-4) + 2(-6) = -5
\]
\[
-3x + 28 - 12 = -5
\]
\[
-3x + 16 = -5
\]
\[
-3x = -21
\]
\[
x = 7
\]

### Solution:

The solution to the system is:
\[
x = 7, \quad y = -4, \quad z = -6
\]

Would you like a deeper explanation or more details about this solution?

Here are five related questions to explore:

1. How would the solution change if one equation were altered slightly?
2. What other methods can be used to solve a system of three variables?
3. How can matrix methods, like Gaussian elimination, be applied here?
4. What are the geometric interpretations of the solution to a system of three equations?
5. How do you check if a solution is correct for a system of linear equations?

**Tip**: Always verify your solution by substituting the values of \(x\), \(y\), and \(z\) back into the original equations to ensure they satisfy all equations.

Solve the system of equations for all three variables:

-3x - 7y + 2z = -5
-3x + 8y - 10z = 7
3x + y + 4z = -7

Learn how to solve a system of linear equations with three variables using elimination and substitution methods. This detailed step-by-step solution covers key algebraic concepts and is suitable for high school students.

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