Solve the system of equations by row reducing the corresponding augmented matrix using matrix...

Question:

Solve the system of equations by row reducing the corresponding augmented matrix using matrix method or the Gauss-Jordan method. Label all row operations

{eq}2x+7y =9{/eq}

{eq}7x +2y =9{/eq}

Gaussian Elimination

We can represent a system of two or more linear equations as an augmented matrix. To do so, we need to construct a matrix containing the coefficients on our variables and then add on one row at the right with the constants after the equals sign. We can then perform elementary row operations with the goal of constructing the identity matrix on the left, and the values in the rightmost row will be the values of the variables.

The augmented matrix consists of the coefficients on x and y, as well as the numbers after the equals sign for each equation.

{eq}\begin{bmatrix} 2 & 7 & 9\\ 7 & 2 & 9\end{bmatrix} {/eq}

If we perform row operations, we can attempt to find the value of each of the two variables. We need to try to construct the identity matrix on the left side of this matrix.

{eq}\begin{align*} &\begin{bmatrix} 2 & 7 & 9\\ 7 & 2 & 9\end{bmatrix} \quad R_1 = \frac{1}{2} R_1\\ &\begin{bmatrix} 1 & 3.5 & 4.5\\ 7 & 2 & 9\end{bmatrix} \quad R_2 = R_2 - 7R_1\\ &\begin{bmatrix} 1 & 3.5 & 4.5\\ 0 & -22.5 & -22.5\end{bmatrix} \quad R_2 = -\frac{1}{22.5} R_2\\ &\begin{bmatrix} 1 & 3.5 & 4.5\\ 0 & 1 & 1 \end{bmatrix} \quad R_1 = R_1 - 3.5R_2\\ &\begin{bmatrix} 1 & 0 & 1\\ 0 & 1 & 1 \end{bmatrix} \end{align*} {/eq}

Therefore, the solution to this equation is {eq}x = 1 {/eq} and {eq}y = 1 {/eq}.