Math Problem Statement

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]].

Solution

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

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.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Gaussian Elimination
Inconsistent Systems

Formulas

Gaussian elimination
Row reduction

Theorems

Inconsistent systems occur when a row of zeros in the coefficient matrix leads to a nonzero value in the augmented column

Suitable Grade Level

Undergraduate level

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