https://file.aiquickdraw.com/mathbot/file/1726607926191yipjyubh.jpg

The system of linear equations from the image is:

\[
\begin{aligned}
1. \quad & x + y + 4z = 17 \\
2. \quad & -3x - 3y - 11z = -48
\end{aligned}
\]

To solve this system, let's use the method of substitution or elimination. I'll first simplify and eliminate one variable. Let's proceed.

### Step 1: Simplify the system
I'll multiply the first equation by 3 to eliminate \(x\) and \(y\) using the second equation.

Multiplying the first equation by 3:

\[
3(x + y + 4z) = 3(17) \quad \Rightarrow \quad 3x + 3y + 12z = 51
\]

Now the system becomes:

\[
\begin{aligned}
3x + 3y + 12z &= 51 \quad \text{(Equation 3)} \\
-3x - 3y - 11z &= -48 \quad \text{(Equation 2)}
\end{aligned}
\]

### Step 2: Add the equations
Now add Equation 2 and Equation 3:

\[
(3x + 3y + 12z) + (-3x - 3y - 11z) = 51 + (-48)
\]

This simplifies to:

\[
z = 3
\]

### Step 3: Substitute \(z = 3\) into the first equation
Now substitute \(z = 3\) into the first equation \(x + y + 4z = 17\):

\[
x + y + 4(3) = 17 \quad \Rightarrow \quad x + y + 12 = 17
\]

This simplifies to:

\[
x + y = 5
\]

### Step 4: Solve for \(x\) and \(y\)
We know \(x + y = 5\), but to uniquely solve for \(x\) and \(y\), more information would typically be needed (such as another equation), or we could express \(x\) in terms of \(y\) (or vice versa). Here, without further information, the solution for the system is:

- \(z = 3\)
- \(x + y = 5\)

Would you like to proceed with further analysis, or do you have any questions about this?

---

### Further Related Questions:
1. How would the solution change if the system included a third equation?
2. Can we use matrix methods like Gaussian elimination to solve this?
3. What is the significance of having infinite solutions in some linear systems?
4. How does the substitution method compare with the elimination method?
5. How can we solve larger systems of equations with more variables?

**Tip**: Always check your solution by substituting it back into the original equations to ensure it satisfies both.

Solve the following system of linear equations:
1. x + y + 4z = 17
2. -3x - 3y - 11z = -48

Learn how to solve the system of linear equations x + y + 4z = 17 and -3x - 3y - 11z = -48 using the elimination method. This problem covers key algebraic concepts and is suitable for students in Grades 9-12.

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