Construct the augmented matrix that corresponds to the following system of equations.

⎧⎩⎨⎪⎪⎪⎪5+7x5=y4z−2(x−2y)=07x−y=2(x−5z)




To construct the augmented matrix for the given system of equations, we first need to rewrite the system in a standard linear form. The standard form is \( ax + by + cz = d \), where \( a \), \( b \), \( c \), and \( d \) are constants. Let's rearrange each equation accordingly:

### System of Equations
1. \( 5 + 7x = y + 4z \)
2. \( -2(x - 2y) = 0 \)
3. \( 7x - y = 2(x - 5z) \)

### Step-by-Step Solution

#### 1. Rearrange the first equation:
   \[
   5 + 7x = y + 4z
   \]
   Subtract \( y + 4z \) from both sides:
   \[
   7x - y - 4z = -5
   \]

#### 2. Simplify the second equation:
   \[
   -2(x - 2y) = 0
   \]
   Distribute the \(-2\):
   \[
   -2x + 4y = 0
   \]

#### 3. Simplify the third equation:
   \[
   7x - y = 2(x - 5z)
   \]
   Distribute the \( 2 \):
   \[
   7x - y = 2x - 10z
   \]
   Subtract \( 2x \) from both sides:
   \[
   5x - y + 10z = 0
   \]

### Final System
\[
\begin{cases}
7x - y - 4z = -5 \\
-2x + 4y = 0 \\
5x - y + 10z = 0
\end{cases}
\]

### Augmented Matrix
The augmented matrix for this system is constructed by aligning the coefficients of \( x \), \( y \), \( z \), and the constants on the right side:

\[
\begin{bmatrix}
7 & -1 & -4 & | & -5 \\
-2 & 4 & 0 & | & 0 \\
5 & -1 & 10 & | & 0
\end{bmatrix}
\]

This matrix represents the coefficients of the variables and the constants from the system of equations.

Would you like more details or have any questions about this process?

Here are 5 related questions:
1. How do you determine if an augmented matrix has a unique solution?
2. What are the steps to perform row reduction on an augmented matrix?
3. How do you find the inverse of a matrix if it exists?
4. What does it mean if an augmented matrix has a row of all zeros?
5. How can an augmented matrix be used to solve a system of equations with more than three variables?

**Tip:** When constructing an augmented matrix, ensure each equation is arranged with the variables aligned and in the same order. This consistency is crucial for accurate matrix formation.

Construct the augmented matrix that corresponds to the following system of equations: 5 + 7x = y + 4z, -2(x - 2y) = 0, 7x - y = 2(x - 5z).

Learn how to construct the augmented matrix for a system of linear equations, including detailed steps to rearrange equations and align coefficients. This guide covers concepts from linear algebra and systems of equations, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation