Math Problem Statement

Construct the augmented matrix that corresponds to the following system of equations: 9 + 2x = y 5z - 4(x - 8y) = 0 4x - y = 2(x - 6z)

Solution

To construct the augmented matrix for this system of equations, we need to first simplify each equation and express them in standard form, ax+by=cax + by = c. Here are the given equations:

  1. 9+2x=y9 + 2x = y
  2. 5z4(x8y)9=0\frac{5z - 4(x - 8y)}{9} = 0
  3. 4xy=2(x6z)4x - y = 2(x - 6z)

Let's rewrite each equation step by step:

Step 1: Simplify the first equation

Rewrite 9+2x=y9 + 2x = y as: 2xy=92x - y = -9

Step 2: Simplify the second equation

Start with: 5z4(x8y)9=0\frac{5z - 4(x - 8y)}{9} = 0 Multiply both sides by 9: 5z4(x8y)=05z - 4(x - 8y) = 0 Expand: 5z4x+32y=05z - 4x + 32y = 0 Rewrite as: 4x+32y+5z=0-4x + 32y + 5z = 0

Step 3: Simplify the third equation

Start with: 4xy=2(x6z)4x - y = 2(x - 6z) Expand the right-hand side: 4xy=2x12z4x - y = 2x - 12z Move terms to one side: 4x2xy+12z=04x - 2x - y + 12z = 0 This simplifies to: 2xy+12z=02x - y + 12z = 0

Step 4: Form the augmented matrix

The system of equations in matrix form is:

  1. 2xy+0z=92x - y + 0z = -9
  2. 4x+32y+5z=0-4x + 32y + 5z = 0
  3. 2xy+12z=02x - y + 12z = 0

The augmented matrix is:

2 & -1 & 0 & | -9 \\ -4 & 32 & 5 & | 0 \\ 2 & -1 & 12 & | 0 \\ \end{bmatrix}$$ Would you like a detailed explanation of any step or have any questions? Here are 5 related questions to consider: 1. How do you convert a system of linear equations into an augmented matrix? 2. What are the steps to solve a system of equations using matrix row reduction? 3. How does changing the form of an equation affect its representation in a matrix? 4. What does each row in an augmented matrix represent geometrically? 5. How can you tell if a system of linear equations is inconsistent using its augmented matrix? **Tip:** When forming augmented matrices, always ensure equations are in standard form $$ax + by + cz = d$$ before constructing the matrix.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
System of Linear Equations
Matrices

Formulas

Ax = b form
Matrix representation of linear systems

Theorems

Matrix Augmentation Theorem
Row Reduction

Suitable Grade Level

Undergraduate (College Level)