Question

Determine the number of solutions for the following system of linear equations. If there is only onesolution, find the solution.- 5x + 2y + 6z = -2x + 3y + 4z = - 36x + y – 2z = -1AnswerKeypadKeyboard ShortcutsSelecting an option will enable input for any required text boxes. If the selected option does not have anyassociated text boxes, then no further input is required.O No SolutionO Only One Solution=こ%3DO Infinitely Many Solutions

Answer 1

We are given the following system of linear equations

$\begin{gathered} -5x+2y+6z=-2 \\ x+3y+4z=-3 \\ 6x+y-2z=-1 \end{gathered}$

Let us solve the given system of equations.

First, let us find the determinant.

The determinant is given by

$\begin{gathered} D=\begin{bmatrix}{-5} & 2 & {6} \\ {1} & {3} & {4} \\ {6} & {1} & {-2}\end{bmatrix}=-5\begin{bmatrix}{3} & {4} \\ 1 & {-2}\end{bmatrix}-2\begin{bmatrix}{1} & {4} \\ {6} & {-2}\end{bmatrix}+6\begin{bmatrix}{1} & {3} \\ {6} & {1}\end{bmatrix} \\ D=\begin{bmatrix}{-5} & 2 & {6} \\ {1} & {3} & {4} \\ {6} & {1} & {-2}\end{bmatrix}=-5(3\cdot-2-4\cdot1)-2(1\cdot-2-4\cdot6)+6(1\cdot1-3\cdot6) \\ D=\begin{bmatrix}{-5} & 2 & {6} \\ {1} & {3} & {4} \\ {6} & {1} & {-2}\end{bmatrix}=-5(-6-4)-2(-2-24)+6(1-18) \\ D=\begin{bmatrix}{-5} & 2 & {6} \\ {1} & {3} & {4} \\ {6} & {1} & {-2}\end{bmatrix}=-5(-10)-2(-26)+6(-17) \\ D=\begin{bmatrix}{-5} & 2 & {6} \\ {1} & {3} & {4} \\ {6} & {1} & {-2}\end{bmatrix}=50+52-102 \\ D=\begin{bmatrix}{-5} & 2 & {6} \\ {1} & {3} & {4} \\ {6} & {1} & {-2}\end{bmatrix}=102-102 \\ D=\begin{bmatrix}{-5} & 2 & {6} \\ {1} & {3} & {4} \\ {6} & {1} & {-2}\end{bmatrix}=0 \end{gathered}$

As you can see, the determinant of the matrix is 0

This means that the given system of equations has no solution.