Matrix Representation of Systems of Linear Equations

Math Problem Statement

Find the matrix corresponding to the system of equations below. Use the same order of the equations in the system for the order of the rows in the matrix.

Solution

The task is to construct the augmented matrix for the given system of linear equations:

$-4x_1 - 8x_2 = 3$ $-9x_1 - 6x_2 + 4x_3 = 8$ $2x_1 + 6x_2 + 7x_3 = -5$

Step-by-step process:

Identify the coefficients and constants:
- First equation: Coefficients are $-4, -8, 0$ (since $x_3$ is missing) and the constant is $3$ .
- Second equation: Coefficients are $-9, -6, 4$ and the constant is $8$ .
- Third equation: Coefficients are $2, 6, 7$ and the constant is $-5$ .
Form the augmented matrix: Each row of the matrix corresponds to the coefficients of the variables $x_1, x_2, x_3$ followed by the constant term.

-4 & -8 & 0 & 3 \\ -9 & -6 & 4 & 8 \\ 2 & 6 & 7 & -5 \end{bmatrix}$$ This is the augmented matrix representation of the system. --- Would you like further clarification or a detailed explanation of any specific part? ### Related Questions: 1. What is the difference between a coefficient matrix and an augmented matrix? 2. How do you solve a system of equations using row-reduction? 3. What are the steps to convert an augmented matrix into reduced row-echelon form? 4. Can this system of equations be solved using the inverse of the coefficient matrix? 5. How can we interpret the solution geometrically? ### Tip: Always double-check the alignment of coefficients with their respective variables to avoid errors when forming the matrix.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Matrix Representation

Formulas

Augmented Matrix Form

Theorems

Matrix Representation of Linear Systems

Suitable Grade Level

Grades 9-12

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