Find the set of solutions for the linear system
4x1 + 6x2 + x3 + x4 = −17
− 2x2 + 8x3 − 12x4 = −5
3x3 + x4 = 12
If your answer includes any arbitrary parameters name
them s1, s2, etc., as needed.

We are given the system of linear equations:

\[
4x_1 + 6x_2 + x_3 + x_4 = -17 \tag{1}
\]
\[
-2x_2 + 8x_3 - 12x_4 = -5 \tag{2}
\]
\[
3x_3 + x_4 = 12 \tag{3}
\]

We will solve this system step by step using substitution and elimination.

### Step 1: Solve equation (3) for \(x_4\).
From equation (3):
\[
x_4 = 12 - 3x_3 \tag{4}
\]

### Step 2: Substitute \(x_4\) into equation (2).
Now substitute equation (4) into equation (2):
\[
-2x_2 + 8x_3 - 12(12 - 3x_3) = -5
\]
Simplifying:
\[
-2x_2 + 8x_3 - 144 + 36x_3 = -5
\]
\[
-2x_2 + 44x_3 - 144 = -5
\]
\[
-2x_2 + 44x_3 = 139 \tag{5}
\]

### Step 3: Substitute \(x_4\) into equation (1).
Now substitute equation (4) into equation (1):
\[
4x_1 + 6x_2 + x_3 + (12 - 3x_3) = -17
\]
Simplifying:
\[
4x_1 + 6x_2 + x_3 + 12 - 3x_3 = -17
\]
\[
4x_1 + 6x_2 - 2x_3 = -29 \tag{6}
\]

### Step 4: Solve for \(x_2\) in terms of \(x_3\).
From equation (5):
\[
-2x_2 + 44x_3 = 139
\]
Solve for \(x_2\):
\[
x_2 = 22x_3 - \frac{139}{2} \tag{7}
\]

### Step 5: Substitute \(x_2\) into equation (6).
Substitute equation (7) into equation (6):
\[
4x_1 + 6(22x_3 - \frac{139}{2}) - 2x_3 = -29
\]
Simplifying:
\[
4x_1 + 132x_3 - 3 \cdot 139 - 2x_3 = -29
\]
\[
4x_1 + 132x_3 - 417 - 2x_3 = -29
\]
\[
4x_1 + 130x_3 = 388 \tag{8}
\]

### Step 6: Solve for \(x_1\) in terms of \(x_3\).
From equation (8):
\[
4x_1 = -130x_3 + 388
\]
\[
x_1 = -\frac{65}{2}x_3 + 97 \tag{9}
\]

### Step 7: General solution.
Now, using equations (4), (7), and (9), we can express all variables in terms of \(x_3\) (which we will call the free parameter \(s_1\)):

\[
x_1 = -\frac{65}{2}s_1 + 97
\]
\[
x_2 = 22s_1 - \frac{139}{2}
\]
\[
x_3 = s_1
\]
\[
x_4 = 12 - 3s_1
\]

### Final Answer:
The set of solutions is:
\[
x_1 = -\frac{65}{2}s_1 + 97
\]
\[
x_2 = 22s_1 - \frac{139}{2}
\]
\[
x_3 = s_1
\]
\[
x_4 = 12 - 3s_1
\]
where \(s_1\) is an arbitrary parameter.

Do you want more details or have any questions?

Here are 5 related questions to extend your understanding:
1. How can you interpret the parameter \(s_1\) geometrically?
2. What would happen if you assigned a specific value to \(s_1\)?
3. How would the solution change if the constants in the equations were modified?
4. What is the geometric nature of the solution set (e.g., line, plane, etc.)?
5. How can you represent the solution set in vector form?

**Tip:** When solving systems of equations, identifying free variables helps you describe the general solution efficiently.

Find the set of solutions for the linear system 4x1 + 6x2 + x3 + x4 = −17, − 2x2 + 8x3 − 12x4 = −5, 3x3 + x4 = 12. If your answer includes any arbitrary parameters, name them s1, s2, etc., as needed.

This detailed solution shows how to solve the system of linear equations 4x1 + 6x2 + x3 + x4 = −17, − 2x2 + 8x3 − 12x4 = −5, 3x3 + x4 = 12 using substitution, elimination, and parametric methods. Learn how to express the solution in terms of a free variable, suitable for advanced high school or early university students.

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