https://file.aiquickdraw.com/mathbot/file/17277468024755ppe3skr.jpg

The problem provided involves an augmented matrix for a system of linear equations. The matrix is:

\[
\begin{pmatrix}
1 & 1 & 3 & | & -1 \\
1 & 2 & -3 & | & -3 \\
7 & 16 & k & | & -24 \\
\end{pmatrix}
\]

### Task: 
Find the value of \( k \) for which the system has **no solutions**.

### Approach:
For the system to have no solutions, it must be **inconsistent**. Inconsistencies typically occur when the matrix is reduced to a form where you get a row like:

\[
0\ 0\ 0\ |\ b \quad \text{(where } b \neq 0)
\]

This indicates that the equations are contradictory.

We can proceed by applying Gaussian elimination to see when such a situation occurs. Let's begin by eliminating the first row from the second and third rows to simplify the matrix. I'll compute this for you.

#### Steps:
1. Subtract Row 1 from Row 2 and Row 3 to simplify the system.
2. Check when a contradiction (inconsistency) arises as a function of \( k \).

I will now compute these steps.It seems there is a placeholder for \( k \) in the third row of the matrix, so I'll substitute \( k \) with a variable and continue with the Gaussian elimination process. Let's apply this correctly to identify where the system becomes inconsistent.

I'll replace \( k \) and continue with the row operations.After performing Gaussian elimination, the matrix is reduced to the following form:

\[
\begin{pmatrix}
1 & 0 & 0 & \frac{k + 24}{k + 33} \\
0 & 1 & 0 & \frac{-2k - 60}{k + 33} \\
0 & 0 & 1 & \frac{1}{k + 33}
\end{pmatrix}
\]

To find when the system has **no solutions**, we need to examine when there is a division by zero, which would make the matrix inconsistent. This occurs when the denominator in any of the terms is zero.

Notice that the denominator is \( k + 33 \) in all terms. For the system to have no solutions, we require \( k + 33 = 0 \), which gives:

\[
k = -33
\]

Thus, the system will have no solutions when \( k = -33 \).

Would you like more details or have any further questions? Here are some related questions:

1. How does Gaussian elimination lead to identifying when a system is inconsistent?
2. What happens if \( k = -33 \) in terms of the row-reduced matrix?
3. Can the system have infinite solutions for any value of \( k \)?
4. How does row reduction help in solving linear systems?
5. How would you handle a different matrix where multiple parameters affect consistency?

**Tip:** Always check for divisions by zero during row reduction, as these are indicators of potential inconsistencies in the system.

For what value of k will the system have no solutions? The augmented matrix is: [[1, 1, 3, -1], [1, 2, -3, -3], [7, 16, k, -24]].

Explore how to find the value of k in the linear system's augmented matrix that results in no solutions using Gaussian elimination. Learn key concepts in linear algebra, including inconsistent systems and row reduction.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation