Solve System of Linear Equations Using Gaussian Elimination: -3x + 8y - z = 2, -2x - 8y - 3z = -8, 5x

Math Problem Statement

−3x+8y−z= 2 minus, 2, x, minus, 8, y, minus, 3, z, equals, minus, 8 −2x−8y−3z= −8 5, x, minus, 8, y, plus, 5, z, equals, 10 5x−8y+5z= 10

The system of linear equations you provided is:

-3x + 8y - z &= 2 \\ -2x - 8y - 3z &= -8 \\ 5x - 8y + 5z &= 10 \end{aligned}$$ We can solve this system using methods like substitution, elimination, or matrix methods (such as Gaussian elimination). I'll proceed with solving this system using Gaussian elimination. ### Step 1: Write the system as an augmented matrix. $$\begin{pmatrix} -3 & 8 & -1 & | & 2 \\ -2 & -8 & -3 & | & -8 \\ 5 & -8 & 5 & | & 10 \end{pmatrix}$$ ### Step 2: Apply Gaussian elimination to solve for $$x$$, $$y$$, and $$z$$. Let me calculate the solution.The solution to the system of equations is: $$x = -2, \quad y \approx 0, \quad z = 4$$ The value of $$y$$ is extremely close to zero, so it can be considered as $$y = 0$$. Would you like more details on any of the steps or methods used in solving this system? Here are 5 related questions you might find interesting: 1. How does Gaussian elimination work for solving systems of equations? 2. What are some alternative methods to solve a system of linear equations? 3. What happens when a system of equations has no solution or infinite solutions? 4. Can matrix operations always guarantee a solution for any system of equations? 5. How does rounding impact the precision of solutions in linear algebra? **Tip:** When working with systems of equations, always check if the system is consistent (has a solution) or inconsistent (no solutions) before solving.

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Linear Algebra
Systems of Linear Equations
Gaussian Elimination

Ax = B (Matrix representation of a system of linear equations)
Gaussian elimination steps to solve systems of linear equations

Gaussian Elimination Theorem
Uniqueness of Solutions Theorem

Grades 9-12

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Solve System of Linear Equations Using Gaussian Elimination