Math Problem Statement

Write the augmented matrix for the system of linear equations: -x - 9y + 8z = 9, -2x - 24z = -35, 3x - y + 9z = 17

Solution

To construct the augmented matrix for this system of linear equations, let’s extract the coefficients from each equation and arrange them in matrix form.

Given system:

-x - 9y + 8z = 9 \\ -2x + 0y - 24z = -35 \\ 3x - y + 9z = 17 \end{cases}$$ For each equation: 1. The coefficients of $$x$$, $$y$$, and $$z$$ in the first equation are $$-1$$, $$-9$$, and $$8$$, with the constant term $$9$$. 2. The coefficients in the second equation are $$-2$$, $$0$$ (since $$y$$ is missing), and $$-24$$, with the constant term $$-35$$. 3. The coefficients in the third equation are $$3$$, $$-1$$, and $$9$$, with the constant term $$17$$. Now, we can write the augmented matrix as: $$\begin{bmatrix} -1 & -9 & 8 & | & 9 \\ -2 & 0 & -24 & | & -35 \\ 3 & -1 & 9 & | & 17 \\ \end{bmatrix}$$ This augmented matrix represents the given system of equations. Would you like further details on matrix operations or solving systems of linear equations? Here are some related questions you might find helpful: 1. How do you solve a system of linear equations using matrix row reduction? 2. What is the difference between an augmented matrix and a coefficient matrix? 3. Can every system of linear equations be solved using an augmented matrix? 4. What are the implications if an augmented matrix has a row of zeros? 5. How does Gaussian elimination work with augmented matrices? **Tip:** When setting up an augmented matrix, always keep track of zero coefficients for missing terms to ensure the matrix structure aligns correctly with the variables.

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Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Linear Equations
Matrix Representation

Formulas

Augmented matrix form for linear equations

Theorems

Linear system representation in matrix form

Suitable Grade Level

High School

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