Use Gauss-Jordan row reduction to solve the given system of equations. (If there is no solution, enter NO SOLUTION. If the system is dependent, express your answer using the parameters x and/or y.)
6x	 − 	7y	 = 	9
12x	 − 	14y	 = 	18
(x, y) = ()

Let's solve this system of linear equations using the Gauss-Jordan elimination method.

The system of equations is:

1. \( 6x - 7y = 9 \)
2. \( 12x - 14y = 18 \)

We will represent this system in augmented matrix form:

\[
\begin{pmatrix}
6 & -7 & | & 9 \\
12 & -14 & | & 18
\end{pmatrix}
\]

### Step 1: First row operation
The second row is a multiple of the first row. Specifically, the second row is obtained by multiplying the first row by 2. To simplify the matrix, we can subtract 2 times the first row from the second row:

\[
R_2 \to R_2 - 2R_1
\]

\[
\begin{pmatrix}
6 & -7 & | & 9 \\
0 & 0 & | & 0
\end{pmatrix}
\]

This leads to the second row being all zeros, indicating that the system is dependent. Thus, we have a system with infinite solutions.

### Step 2: Solve for one variable in terms of the other
From the first row:

\[
6x - 7y = 9
\]

Solving for \( x \):

\[
x = \frac{9 + 7y}{6}
\]

Thus, the solution set is:

\[
x = \frac{9 + 7y}{6}, \quad y = y
\]

This is the parametric form of the solution, where \( y \) is a free parameter.

### Final Answer:
\[
(x, y) = \left( \frac{9 + 7y}{6}, y \right)
\]

Would you like more details or have any questions?

### Related questions:
1. How do you apply row reduction to larger systems?
2. What are the steps to identify a dependent system?
3. Can row reduction always be used to find solutions to linear systems?
4. How do you interpret an augmented matrix with all-zero rows?
5. How do you check if a system has no solution?

**Tip**: In a dependent system, any free variables can be assigned arbitrary values to generate infinitely many solutions.

Use Gauss-Jordan row reduction to solve the given system of equations. (If there is no solution, enter NO SOLUTION. If the system is dependent, express your answer using the parameters x and/or y.) 6x − 7y = 9, 12x − 14y = 18 (x, y) = ()

Learn how to solve the system of equations 6x − 7y = 9, 12x − 14y = 18 using Gauss-Jordan elimination. This problem demonstrates key linear algebra concepts, including dependent systems and parametric solutions. Suitable for high school 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