Math Problem Statement

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

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

Solution

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

Given:

  1. The system equation is:
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.

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Math Problem Analysis

Mathematical Concepts

Linear Systems
Vector Equations
Underdetermined Systems
Parametric Solutions

Formulas

System of linear equations: x1 + x2 + x3 + x4 = c
Parametric form: X = X0 + s * D

Theorems

Inconsistent Systems Theorem
Parametric Solution Theorem

Suitable Grade Level

Grades 10-12

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