Math Problem Statement

Solve the system of equations: \[ \begin{cases} x_1 + x_2 = 5 \\ x_2 + x_3 = -3 \\ x_3 + x_4 = 2 \\ x_1 + x_4 = 10 \end{cases} \]

Solution

To solve the system of linear equations given in the image, the system can be written as follows:

x_1 + x_2 = 5 \\ x_2 + x_3 = -3 \\ x_3 + x_4 = 2 \\ x_1 + x_4 = 10 \end{cases}$$ --- ### Step-by-Step Solution #### 1. Represent the system. We write it as a matrix equation $$A \mathbf{x} = \mathbf{b}$$, where: $$A = \begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 5 \\ -3 \\ 2 \\ 10 \end{bmatrix}.$$ The system becomes: $$\begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} = \begin{bmatrix} 5 \\ -3 \\ 2 \\ 10 \end{bmatrix}.$$ --- #### 2. Solve the system step by step. **Equation 1:** $$x_1 + x_2 = 5$$. We express $$x_1$$ in terms of $$x_2$$: $$x_1 = 5 - x_2.$$ **Equation 2:** $$x_2 + x_3 = -3$$. We express $$x_3$$ in terms of $$x_2$$: $$x_3 = -3 - x_2.$$ **Equation 3:** $$x_3 + x_4 = 2$$. Substitute $$x_3 = -3 - x_2$$: $$(-3 - x_2) + x_4 = 2.$$ Simplify for $$x_4$$: $$x_4 = 5 + x_2.$$ **Equation 4:** $$x_1 + x_4 = 10$$. Substitute $$x_1 = 5 - x_2$$ and $$x_4 = 5 + x_2$$: $$(5 - x_2) + (5 + x_2) = 10.$$ Simplify: $$10 = 10.$$ This equation is consistent, so no contradictions arise. --- #### 3. General solution. From the above steps: - $$x_1 = 5 - x_2$$, - $$x_3 = -3 - x_2$$, - $$x_4 = 5 + x_2$$. Thus, the solution is: $$\mathbf{x} = \begin{bmatrix} 5 - x_2 \\ x_2 \\ -3 - x_2 \\ 5 + x_2 \end{bmatrix},$$ where $$x_2$$ is a free parameter. --- ### Final Answer: The solution can be written as: $$\mathbf{x} = \begin{bmatrix} 5 \\ 0 \\ -3 \\ 5 \end{bmatrix} + s \begin{bmatrix} -1 \\ 1 \\ -1 \\ 1 \end{bmatrix},$$ where $$s = x_2$$ is a free parameter. --- ### Do you need further explanation or have any questions? --- ### Related Questions: 1. What is the general solution of a system of linear equations with free variables? 2. How do we determine if a system of equations has infinitely many solutions? 3. What is the geometric interpretation of this solution in 4D space? 4. How can this problem be solved using row reduction (Gaussian elimination)? 5. How does the rank of the coefficient matrix determine the type of solutions? --- **Tip:** When solving systems of equations, express variables in terms of free variables to describe the solution set fully.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Equations
General Solution with Free Variables

Formulas

Matrix form: A * x = b
Solution involving free variables

Theorems

Consistency of Linear Systems
General Solution of Linear Equations

Suitable Grade Level

Undergraduate or Advanced High School (Grades 11-12)