Math Problem Statement

Solve the system of equations: -14x + 7y - 42z = 42, -9x + 4y - 32z = 25, -2x + y - 6z = 6.

Solution

Let's analyze and solve the given system of equations step by step.

The system is:

-14x + 7y - 42z &= 42, \\ -9x + 4y - 32z &= 25, \\ -2x + y - 6z &= 6. \end{aligned}$$ ### Step 1: Write the system in augmented matrix form $$\begin{bmatrix} -14 & 7 & -42 & | & 42 \\ -9 & 4 & -32 & | & 25 \\ -2 & 1 & -6 & | & 6 \end{bmatrix}.$$ ### Step 2: Perform row reduction to determine the nature of the system We aim to reduce the augmented matrix to row echelon form to see if the system is dependent or inconsistent. 1. Divide the first row by $$-14$$ to simplify: $$\begin{bmatrix} 1 & -\frac{1}{2} & 3 & | & -3 \\ -9 & 4 & -32 & | & 25 \\ -2 & 1 & -6 & | & 6 \end{bmatrix}.$$ 2. Use the first row to eliminate the first variable ($$x$$) from rows 2 and 3: - Row 2: $$\text{Row 2} + 9 \times \text{Row 1}$$, - Row 3: $$\text{Row 3} + 2 \times \text{Row 1}$$. After performing these operations: $$\begin{bmatrix} 1 & -\frac{1}{2} & 3 & | & -3 \\ 0 & -\frac{1}{2} & -5 & | & -2 \\ 0 & 0 & 0 & | & 0 \end{bmatrix}.$$ ### Step 3: Interpret the results - The last row is $$0 = 0$$, which is true and does not introduce a contradiction. This indicates that the system might be **dependent**. - The second row simplifies further. Dividing it by $$-\frac{1}{2}$$: $$\begin{bmatrix} 1 & -\frac{1}{2} & 3 & | & -3 \\ 0 & 1 & 10 & | & 4 \\ 0 & 0 & 0 & | & 0 \end{bmatrix}.$$ ### Step 4: Express the solution From the row-reduced form: 1. $$x - \frac{1}{2}y + 3z = -3 \quad \Rightarrow \quad x = \frac{1}{2}y - 3z - 3.$$ 2. $$y + 10z = 4 \quad \Rightarrow \quad y = -10z + 4.$$ Substitute $$y = -10z + 4$$ into $$x = \frac{1}{2}y - 3z - 3$$: $$x = \frac{1}{2}(-10z + 4) - 3z - 3 = -5z + 2 - 3z - 3 = -8z - 1.$$ ### Final Answer: The system is **dependent**, and the solution is: $$x = -8z - 1, \quad y = -10z + 4.$$ Would you like a detailed explanation of any step or further assistance? --- Here are 5 related questions to deepen your understanding: 1. What does it mean for a system of equations to be dependent or inconsistent? 2. How do you interpret the augmented matrix during row reduction? 3. What are the key differences between row echelon form and reduced row echelon form? 4. How can the solution $$x, y, z$$ be verified by substituting back into the original equations? 5. How can parametric solutions like this one be visualized geometrically? **Tip:** Always check for errors in row reduction by substituting the solutions back into the original equations!

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Math Problem Analysis

Mathematical Concepts

Systems of Linear Equations
Row Reduction
Augmented Matrices
Parametric Solutions

Formulas

Row operations for augmented matrix
x = 1/2 y - 3z - 3
y = -10z + 4

Theorems

Row Echelon Form
Dependent Systems of Equations

Suitable Grade Level

Grades 9-11

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