Chapter 8, Problem 19CCSR

Solution

To find: The $y-$ coefficient of the first equation of the system where the matrix $\left[\begin{array}{cc}3& -1\\ 1& 2\end{array}\left|\begin{array}{c}5\\ -1\end{array}\right.\right]$ represents a linear system of equations.

The $y-$ coefficient of the first equation of the system of equation is $-1$ .

Given:

The matrix representing the system of equations is $\left[\begin{array}{cc}3& -1\\ 1& 2\end{array}\left|\begin{array}{c}5\\ -1\end{array}\right.\right]$ .

Concept used:

The system of linear equations $a_{1}x+b_{1}y=c_1\text{ and }{a}_{2}x+b_{2}y=c_2$ in augmented matrix are written as $\left[\begin{array}{cc}{a}_1& b_1\\ a_2& b_2\end{array}\left|\begin{array}{c}{c}_1\\ c_2\end{array}\right.\right]$ where the vertical bar in augmented matrix represents the equality sign for system of equations and first row represents the first equation and second row represents the second equation.

Calculation:

With the help of concept stated above, the system of equation from the augmented matrix $\left[\begin{array}{cc}3& -1\\ 1& 2\end{array}\left|\begin{array}{c}5\\ -1\end{array}\right.\right]$ is written as $3x-y=5\text{ and }x-2y=-1$ .

Therefore, the $y-$ coefficient of the first equation of the system of equation is $-1$ .

Conclusion:

The $y-$ coefficient of the first equation of the system of equation is $-1$ .