Solving 3x3 Systems of Equations using Augmented Matrices

Math Problem Statement

find the augmented matrix and rref matrix and solution for each problem

Find the augmented matrix and RREF matrix, and solution for each problem in the 3x3 systems of equations worksheet.

Solution

Let's solve each system of equations by representing it in an augmented matrix form and then finding its reduced row echelon form (RREF). I'll begin by breaking down each problem.

Problem 7

Equations:

$2x + 6y + z = 5$
$x + y - z + 1 = 0$ → rewrite as $x + y - z = -1$
$x + 8y + 2z = 0$

Augmented Matrix Form:

2 & 6 & 1 & | & 5 \\ 1 & 1 & -1 & | & -1 \\ 1 & 8 & 2 & | & 0 \end{bmatrix}$$ ### Finding the RREF: Now, let's solve the augmented matrix using row operations to obtain the RREF. 1. **Row 1: $$[2, 6, 1 | 5]$$ → Divide by 2:** $$[1, 3, 0.5 | 2.5]$$ 2. **Make the leading 1 in the second row:** - Row 2 is already leading with $$x + y - z = -1$$. - We need to make $$R2 = [0, -7, -1.5 | -3.5]$$

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Equations
Matrix Operations

Formulas

Ax = B (where A is the coefficient matrix, x is the variable matrix, and B is the constant matrix)
Row reduction to RREF (Reduced Row Echelon Form)

Theorems

Gauss-Jordan elimination
Matrix Inversion for solving systems of equations

Suitable Grade Level

Grades 10-12

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