Find all values of k for which the given augmented matrix corresponds to a consistent linear system. (a) [3 -4 k, -6 8 5]

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Find all values of k for which the given augmented matrix corresponds to a consistent linear system.  (a) [3 -4 k, -6 8 5]

Pulad1971 · Accepted Answer

Consider the augmented matrix  [3−4k−685]  Adding 2 times the first row of the matrix above to the second row we obtain  [3−4k002k+5]  which corresponds to the simplified linear system  3x−4y=k  0=2k+5  If k=−52 the second equation holds and then the solutions of the system  3x−4y=k  0=2k+5  are the ones for which x and y satisfy the single equation 3x−4y=−52  If k≠−52 the second equation does not hold and therefore the system is inconsistent.  Hence a linear system with corresponding augmented matrix  [3−4k−685]  is consistent if and only if k=−52

nick1337 · Accepted Answer

Step 1:Starting with the augmented matrix:[3−4k−685]We can perform row operations to simplify the matrix. Using elementary row operations, we aim to obtain a row of zeros or a row with all zeros on the left side.First, let&#039;s multiply the first row by 2 and add it to the second row:[3−4k005+2k]Now, let&#039;s divide the second row by 2:[3−4k005+2k2]Step 2:Simplifying the matrix further, we have:[3−4k005+2k2]To determine the consistency of the system, we examine the echelon form of the matrix. Since the second row has a leading coefficient of 0 and a non-zero constant term, the system will only be consistent if the expression 5+2k2 equals zero.Therefore, to find the values of k for which the augmented matrix corresponds to a consistent linear system, we solve the equation:5+2k2=0To do that, let&#039;s multiply both sides of the equation by 2:5+2k=0Now, subtract 5 from both sides:2k=−5Finally, divide both sides by 2:k=−52Hence, the value of k for which the given augmented matrix corresponds to a consistent linear system is k=−52.

Eliza Beth13 · Accepted Answer

Result:k=−52Solution:Given the augmented matrix:[3−4k−685]We can start the row reduction process. Our goal is to obtain a row echelon form that looks like this:[1*a**b*01*c*]where a, b, and c are constants. Let&#039;s perform the row reduction:R2←R2+2R1This operation gives us the new matrix:[3−4k005+2k]Now, we can see that the second row is of the form [0 0 | *c*] with c = 5 + 2k. To have a consistent linear system, c must be equal to zero. Therefore:5+2k=0Solving for k, we find:2k=−5k=−52Hence, the value of k that makes the augmented matrix correspond to a consistent linear system is k=−52.

Don Sumner · Accepted Answer

To determine the values of k for which the given augmented matrix corresponds to a consistent linear system, we need to perform row operations and analyze the resulting matrix.Let&#039;s denote the augmented matrix as A:A=[3−4k−685]We&#039;ll use elementary row operations to simplify the matrix and bring it into row-echelon form or reduced row-echelon form.First, let&#039;s perform the row operation: R2→R2+2R1 to eliminate the coefficient in the first column:A=[3−4k005+2k]Now, let&#039;s analyze the resulting matrix:If 5+2k≠0, then the second row will have a pivot position (a leading 1), and the system will have a unique solution. Therefore, for k≠−52, the system is consistent.If 5+2k=0, then the second row will have no pivot position, and the system will have infinitely many solutions. Therefore, for k=−52, the system is consistent.To summarize:The given augmented matrix corresponds to a consistent linear system for the following values of k:k=−52For all other values of k, the system is also consistent.

Find all values of k for which the given augmented

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