Math Problem Statement

Compute all solutions of the following linear equation system: x + y + z = 0 x + 2y - z = 0 x - y + 6z = 1

Solution

We are given the following system of linear equations:

1) & \quad x + y + z = 0 \\ 2) & \quad x + 2y - z = 0 \\ 3) & \quad x - y + 6z = 1 \end{aligned}$$ ### Step 1: Use equation (1) to solve for $$x$$ From equation (1): $$x + y + z = 0 \implies x = -y - z$$ Substitute this into equations (2) and (3). ### Step 2: Substitute $$x = -y - z$$ into equation (2) Substitute into equation (2): $$(-y - z) + 2y - z = 0$$ Simplify: $$-y - z + 2y - z = 0 \implies y - 2z = 0$$ Thus, we have: $$y = 2z$$ ### Step 3: Substitute $$x = -y - z$$ and $$y = 2z$$ into equation (3) Now, substitute $$y = 2z$$ and $$x = -y - z = -2z - z = -3z$$ into equation (3): $$(-3z) - (2z) + 6z = 1$$ Simplify: $$-3z - 2z + 6z = 1 \implies z = 1$$ ### Step 4: Solve for $$y$$ and $$x$$ Now that we know $$z = 1$$, substitute this into $$y = 2z$$ to find $$y$$: $$y = 2(1) = 2$$ Substitute $$z = 1$$ and $$y = 2$$ into $$x = -y - z$$ to find $$x$$: $$x = -(2) - 1 = -3$$ ### Final Solution The solution to the system of equations is: $$x = -3, \quad y = 2, \quad z = 1$$ Let me know if you'd like more details on any step, and here are some related questions: 1. How do you solve a system of linear equations using matrix methods like Gaussian elimination? 2. What is Cramer's Rule and how does it apply to 3x3 systems of equations? 3. How do parametric solutions differ from exact solutions in systems of equations? 4. How can systems of linear equations be solved using substitution or elimination? 5. What are the conditions for a system of equations to have no solution or infinitely many solutions? **Tip**: Always verify your solutions by plugging the values back into the original system of equations to check for consistency.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations

Formulas

Substitution method
System of equations solving technique

Theorems

Basic linear systems properties
Unique solution theorem

Suitable Grade Level

College-level Algebra or High School Advanced Algebra

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