Solving a System of Linear Equations with Three Variables

Math Problem Statement

The system of equations is: -2x - 6y - 9z = 5 -2x - 5y - 8z = 5 5x + 15y + 25z = 1

The system of equations provided in the image is:

-2x - 6y - 9z = 5 \\ -2x - 5y - 8z = 5 \\ 5x + 15y + 25z = 1 \end{cases}$$ We need to solve this system for $$x$$, $$y$$, and $$z$$. Let me compute the solution by solving this system of equations.The solution to the system of equations is: $$x = -\frac{53}{5}, \quad y = -\frac{27}{5}, \quad z = \frac{27}{5}$$ Would you like further details or have any questions? Here are 5 related questions: 1. How do you determine if a system of equations has a unique solution? 2. What methods can be used to solve systems of linear equations? 3. How can you verify if a solution satisfies all equations in a system? 4. Can a system of linear equations have no solution or infinitely many solutions? 5. How does substitution differ from elimination in solving linear systems? **Tip:** When solving systems, try using matrix methods like Gaussian elimination for larger systems.

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Linear Algebra
Systems of Equations
Matrix Methods

Ax = b (where A is a matrix of coefficients, x is the vector of variables, and b is the constants vector)
Gaussian Elimination
Substitution

Unique Solution Theorem for Systems of Equations

Grades 10-12

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Solving a System of Linear Equations using Gaussian Elimination