Math Problem Statement

Solve the system of equations: x - y - z = 2, -x + 2y - 3z = -5, 3x - y - 11z = 0. If the system has no solution, state that it is inconsistent.

Solution

Let's solve the system of equations to determine the correct answer.

The system of equations is:

1. \quad x - y - z &= 2 \\ 2. \quad -x + 2y - 3z &= -5 \\ 3. \quad 3x - y - 11z &= 0 \end{align*}$$ ### Step 1: Add equations (1) and (2) to eliminate $$x$$: From equation (1): $$x - y - z = 2$$ From equation (2): $$-x + 2y - 3z = -5$$ Add these two: $$(x - y - z) + (-x + 2y - 3z) = 2 + (-5)$$ $$y - 4z = -3 \quad \text{(Equation 4)}$$ ### Step 2: Add equation (1) and (3) to eliminate $$x$$: From equation (1): $$x - y - z = 2$$ From equation (3): $$3x - y - 11z = 0$$ Multiply equation (1) by 3 and then subtract equation (3): $$(3x - 3y - 3z) - (3x - y - 11z) = 6 - 0$$ $$-2y + 8z = 6 \quad \text{(Equation 5)}$$ ### Step 3: Solve the system of two equations in $$y$$ and $$z$$: Now we have two equations: $$\begin{align*} 4. \quad y - 4z &= -3 \\ 5. \quad -2y + 8z &= 6 \end{align*}$$ Solve equation (4) for $$y$$: $$y = 4z - 3$$ Substitute this into equation (5): $$-2(4z - 3) + 8z = 6$$ $$-8z + 6 + 8z = 6$$ $$6 = 6$$ This is a true statement, meaning the system has infinitely many solutions. Therefore, the system is **not inconsistent**. ### Step 4: Parametrize the solutions: Since there are infinitely many solutions, we parametrize $$z$$ as a free variable. From equation (4), we have $$y = 4z - 3$$. Substitute into equation (1) to find $$x$$: $$x - (4z - 3) - z = 2$$ $$x - 4z + 3 - z = 2$$ $$x = 5z - 1$$ ### Final Solution: The solution set is: $$x = 5z - 1, \quad y = 4z - 3, \quad z = z$$ This matches choice **B** in the question. Would you like more details on any step, or have any further questions? ### 5 Relative Questions: 1. How can we solve a system of linear equations using matrices? 2. What are the different types of solutions for a system of linear equations? 3. How do we know if a system has no solution (inconsistent)? 4. Can all systems with infinitely many solutions be written in parametric form? 5. What is Gaussian elimination, and how does it help in solving linear systems? ### Tip: When solving systems of equations, always check for consistency early to avoid unnecessary calculations.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Parametric Solutions

Formulas

Linear equation: ax + by + cz = d

Theorems

Consistency in linear systems

Suitable Grade Level

Grades 10-12

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