Solve System of Linear Equations with Given Solution (x=1, y=-2, z=2)

Math Problem Statement

Solve the system of linear equations: \(-x + 3y - z = -9\), \(2x + 3z = 8\), \(5x + 2y = 1\). Solution given: \((x, y, z) = (1, -2, 2)\)

The system of equations shown in the image is:

-x + 3y - z = -9, \\ 2x + 3z = 8, \\ 5x + 2y = 1. \end{cases}$$ The solution provided in the image is $$(x, y, z) = (1, -2, 2)$$. ### Solution Verification Let's substitute $$x = 1$$, $$y = -2$$, and $$z = 2$$ into each equation to verify if this solution is correct: 1. **First equation:** $$-x + 3y - z = -9$$ Substitute $$x = 1$$, $$y = -2$$, and $$z = 2$$: $$-1 + 3(-2) - 2 = -1 - 6 - 2 = -9.$$ This holds true. 2. **Second equation:** $$2x + 3z = 8$$ Substitute $$x = 1$$ and $$z = 2$$: $$2(1) + 3(2) = 2 + 6 = 8.$$ This is also correct. 3. **Third equation:** $$5x + 2y = 1$$ Substitute $$x = 1$$ and $$y = -2$$: $$5(1) + 2(-2) = 5 - 4 = 1.$$ This is correct as well. Since all three equations hold with $$(x, y, z) = (1, -2, 2)$$, the provided solution is indeed correct. Would you like further details on solving this system using a particular method, or do you have any questions? ### Related Questions 1. How would you solve this system using matrix methods (Gaussian elimination or Cramer's rule)? 2. Can we solve this system by substitution, and how would that work? 3. What are the possible methods to solve systems of linear equations with three variables? 4. How can we interpret this solution geometrically? 5. How can we verify solutions of larger systems of equations? **Tip:** When solving systems of equations, always verify by plugging the values back into the original equations to ensure consistency.

Drop file here or Click Here to upload

Algebra
Systems of Linear Equations
Substitution
Elimination

Solution of a system of linear equations using substitution or elimination

Consistency and uniqueness of solutions in systems of linear equations

Grades 10-12

Solve the System of Linear Equations: 2x + y - z = -1, x + 2y + z = 4, 3x - y + z = -9

Solve a System of Three Linear Equations in Three Variables

Solve System of Equations: 2x + 4y + z = 9, -2x + 10y + 3z = -1, -2x + 7y + 3z = 5

Solving a System of Linear Equations: Three Variables

Solve a System of Equations with Three Variables