Use Cramer's rule to solve the system or to determine that the system is inconsistent or contains...

Question:

Use Cramer's rule to solve the system or to determine that the system is inconsistent or contains dependent equations:

x + y = 9

x - y = 1.

System of Equations:

Let us assume that we have a {eq}3\times 3 {/eq} system of equations. To find the solution of the system of equations by Cramer's Rule we find the determinant {eq}D, {/eq} by using the the coefficients {eq}x,y {/eq} and {eq}z {/eq} values from the problem. Find the determinant {eq}D_{x}, {/eq} by replacing the {eq}x {/eq}-values in the first column leaving the {eq}y {/eq} and {eq}z {/eq} columns unchanged. Similarly we find the values of {eq}D_{y} {/eq} and {eq}D_{z}. {/eq} Then the solution of the system of equations by Cramer's Rule is as follows

{eq}\displaystyle x=\frac{D_{x}}{D}\\ \displaystyle y=\frac{D_{y}}{D}\\ \displaystyle z=\frac{D_{z}}{D} {/eq}

Consider the system of equations

{eq}\displaystyle x + y = 9\\ \displaystyle x - y = 1 {/eq}

Rewrite the system of equations in matrix form

{eq}\displaystyle \begin{bmatrix} 1 &1 \\ 1&- 1 \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix}=\begin{bmatrix} 9\\ 1 \end{bmatrix} {/eq}

First, find the determinant of the coefficient matrix:

{eq}\displaystyle D=\begin{vmatrix} 1 &1 \\ 1&- 1 \end{vmatrix}\\ \displaystyle =-1-1\\ \displaystyle =-2 {/eq}

{eq}D_{x}: {/eq} coefficient determinant with answer-column values in x-colum

{eq}\displaystyle D_{x} =\begin{vmatrix}9&1\\ 1&-1\end{vmatrix}\\ \displaystyle =-9-1\\ \displaystyle =-10 {/eq}

{eq}\displaystyle D_{y}:{/eq} coefficient determinant with answer-column values in y-colum

{eq}\displaystyle D_{y} =\begin{vmatrix}1&9\\ 1&1\end{vmatrix} \displaystyle =1-9\\ \displaystyle =-8 {/eq}

Solve by using Cramer Rule

{eq}\displaystyle x=\frac{D_x}{D},\:y=\frac{D_y}{D} {/eq}

D denotes the determinant

{eq}\displaystyle x=\frac{D_x}{D}=\frac{-10}{-2}=5\\ \displaystyle y=\frac{D_y}{D}=\frac{-8}{-2}=4 {/eq}

Hence, system is consistence and have an unique solution {eq}\displaystyle x=5, \,\, y=4 {/eq}