Solve the system
⎧⎩⎨⎪⎪⎪⎪⎪⎪x1x1+x2x2+x3x3+x4+x4====−55−4−14

⎡⎣⎢⎢⎢x1x2x3x4⎤⎦⎥⎥⎥=
 ⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
-14
9
-4
0
 ⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
 +s
 ⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
-1
1
-1
1
 ⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
.

The given system involves a vector equation. Let's break it down carefully and solve for the solution.

### Given:
1. The system equation is:

\[
\begin{aligned}
x_1 + x_2 + x_3 + x_4 &= -5 \\
x_1 + x_2 + x_3 + x_4 &= -4 \\
x_1 + x_2 + x_3 + x_4 &= -14
\end{aligned}
\]

This appears to be a system of three equations in four variables, which generally indicates an underdetermined system (there are infinitely many solutions).

2. You are also given:

\[
\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix}
=
\begin{bmatrix}
-14 \\
9 \\
-4 \\
0
\end{bmatrix}
+ s
\begin{bmatrix}
-1 \\
1 \\
-1 \\
1
\end{bmatrix}
\]

Here, \( s \) is a scalar parameter.

### Step 1: Analyze the system of equations

From the system:
\[
x_1 + x_2 + x_3 + x_4 = -5
\]
\[
x_1 + x_2 + x_3 + x_4 = -4
\]
\[
x_1 + x_2 + x_3 + x_4 = -14
\]

These equations are inconsistent because the left-hand sides are all identical (i.e., \( x_1 + x_2 + x_3 + x_4 \)) but the right-hand sides are different (i.e., \(-5\), \(-4\), and \(-14\)).

Therefore, this system has no solution unless there was an error in the problem formulation.

### Step 2: Parametric form

The vector equation provided:

\[
\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix}
=
\begin{bmatrix}
-14 \\
9 \\
-4 \\
0
\end{bmatrix}
+ s
\begin{bmatrix}
-1 \\
1 \\
-1 \\
1
\end{bmatrix}
\]

represents the general solution in parametric form for \( x_1, x_2, x_3, x_4 \), where \( s \) is the parameter. So, any value of \( s \) will give a different solution.

### Conclusion:

- The system of equations is inconsistent, as mentioned earlier, unless there's a modification.
- The vector equation represents a family of solutions in terms of the parameter \( s \).

Would you like me to clarify or further expand on any of the steps?

### Here are 5 questions that extend this topic:
1. How do we interpret an inconsistent system geometrically?
2. What is the significance of having an underdetermined system of equations?
3. How can we find the value of \( s \) if additional conditions are given?
4. What does the parametric solution tell us about the nature of the system?
5. How do you determine if a system has infinitely many solutions or no solution?

#### Tip: 
When dealing with systems of equations, always check for consistency first by comparing the structure of the equations.

Solve the system ⎧⎩⎨⎪⎪⎪⎪⎪⎪x1x1+x2x2+x3x3+x4+x4====−55−4−14 ⎡⎣⎢⎢⎢x1x2x3x4⎤⎦⎥⎥⎥= ⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢ -14 9 -4 0 ⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥ +s ⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢ -1 1 -1 1 ⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥.

This problem involves solving a system of linear equations, highlighting parametric solutions and inconsistent systems. Explore the solution to the vector equation with detailed breakdowns, formulas, and steps. Suitable for Grades 10-12.

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