Math Problem Statement

Solve the linear system of equations: -3x1 + 3x2 - 2x3 + 3x4 = 6, -3x1 - 3x2 - 3x3 = 6, x4 = -1.

Solution

The image contains a linear system of equations presented in matrix form. The system is as follows:

-3x_1 + 3x_2 - 2x_3 + 3x_4 = 6 \\ -3x_1 - 3x_2 - 3x_3 = 6 \\ x_4 = -1 \end{cases}$$ ### Step-by-Step Solution: 1. **From the third equation**, we directly have $$x_4 = -1$$. 2. **Substitute $$x_4 = -1$$ into the first and second equations:** - For the **first equation**: $$-3x_1 + 3x_2 - 2x_3 + 3(-1) = 6 \implies -3x_1 + 3x_2 - 2x_3 - 3 = 6 \implies -3x_1 + 3x_2 - 2x_3 = 9$$ Simplifying by dividing by 3: $$-x_1 + x_2 - \frac{2}{3}x_3 = 3$$ - For the **second equation**: $$-3x_1 - 3x_2 - 3x_3 = 6$$ Simplifying by dividing by -3: $$x_1 + x_2 + x_3 = -2$$ 3. **Now we have a simplified system of two equations:** $$\begin{cases} -x_1 + x_2 - \frac{2}{3}x_3 = 3 \\ x_1 + x_2 + x_3 = -2 \end{cases}$$ 4. **Adding the two equations** to eliminate $$x_1$$: $$(-x_1 + x_2 - \frac{2}{3}x_3) + (x_1 + x_2 + x_3) = 3 - 2$$ $$2x_2 + \frac{1}{3}x_3 = 1$$ Multiply by 3 to clear the fraction: $$6x_2 + x_3 = 3 \implies x_3 = 3 - 6x_2$$ 5. **Substitute $$x_3 = 3 - 6x_2$$ back into the second equation:** $$x_1 + x_2 + (3 - 6x_2) = -2$$ $$x_1 + x_2 - 6x_2 + 3 = -2$$ $$x_1 - 5x_2 = -5$$ $$x_1 = -5 + 5x_2$$ ### Final Solution: The general solution in terms of the parameter $$x_2$$ is: $$\begin{cases} x_1 = 5x_2 - 5 \\ x_2 = x_2 \\ x_3 = 3 - 6x_2 \\ x_4 = -1 \end{cases}$$ ### Parameter Form: If we let $$x_2 = t$$, the solution becomes: $$\begin{cases} x_1 = 5t - 5 \\ x_2 = t \\ x_3 = 3 - 6t \\ x_4 = -1 \end{cases}$$ ### Do you want more details on a specific step or have any questions? ### Related Questions: 1. How can I solve linear systems using matrix operations? 2. What are the steps to convert a system into row echelon form? 3. How do parameterized solutions work in linear algebra? 4. Can we graphically interpret this solution set? 5. What are some practical applications of linear systems? ### Tip: Always check your solution by substituting back into the original equations to verify correctness.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Equations
Parameterization

Formulas

Substitution Method
Simplifying Linear Equations
Parameterization of Solutions

Theorems

System of Linear Equations
Row Reduction
Gaussian Elimination

Suitable Grade Level

College level or advanced high school level

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