Math Problem Statement

Solución del problema

Solution

To solve the problem in the image, let's analyze the system of equations presented:

The problem asks for the values of mm such that the following system has a solution:

x + y + z &= 5 \quad \text{(Equation 1)}\\ x - y + z &= m \quad \text{(Equation 2)}\\ 2x + y - z &= 3 \quad \text{(Equation 3)} \end{aligned}$$ **Step 1: Add Equation 1 and Equation 2.** $$(x + y + z) + (x - y + z) = 5 + m$$ This simplifies to: $$2x + 2z = 5 + m$$ $$x + z = \frac{5 + m}{2} \quad \text{(Equation 4)}$$ **Step 2: Add Equation 3 to Equation 4.** $$(2x + y - z) + (x + z) = 3 + \frac{5 + m}{2}$$ This simplifies to: $$3x + y = 3 + \frac{5 + m}{2}$$ **Step 3: Find $$y$$ in terms of $$x$$ and $$m$$.** Rearrange the equation: $$y = 3 + \frac{5 + m}{2} - 3x$$ **Step 4: Plug $$y$$ back into Equation 1.** Substitute $$y = 3 + \frac{5 + m}{2} - 3x$$ into Equation 1: $$x + \left(3 + \frac{5 + m}{2} - 3x\right) + z = 5$$ This simplifies to: $$\frac{11 + m}{2} - 2x + z = 2$$ Simplifying further, you get: $$z = 2x - \frac{11 + m}{2} + 2$$ **Step 5: Plug $$z$$ back into Equation 4.** Substitute $$z = 2x - \frac{11 + m}{2} + 2$$ into Equation 4: $$x + 2x - \frac{11 + m}{2} + 2 = \frac{5 + m}{2}$$ Simplifying: $$3x = \frac{5 + m}{2} + \frac{11 + m}{2} - 2$$ Simplify to: $$3x = \frac{16 + 2m}{2} - 2$$ Solve for $$x$$: $$x = \frac{8 + m}{3}$$ To ensure the system admits a solution, the corresponding values of $$m$$ must satisfy the conditions imposed by each equation. Checking the given choices, you find that $$m = 1$$ is a valid solution, corresponding to Option **C) (1, -9, 5)**. Thus, the correct answer is **C) (1, -9, 5)**. Would you like more details or have any questions? Here are five related questions to consider: 1. What happens if the system has no solution? 2. How can you determine if a system is inconsistent? 3. What are the steps to find the solution to a system of three linear equations? 4. How does the value of $$m$$ affect the solutions of this system? 5. Can you generalize the solution method for any similar system? **Tip:** When solving systems of equations, always look for opportunities to simplify by adding or subtracting equations to eliminate variables.

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Math Problem Analysis

Mathematical Concepts

Systems of linear equations
Matrix algebra
Solution methods for systems of equations

Formulas

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Theorems

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Suitable Grade Level

Grades 10-12

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