Chapter 16.9, Problem 2WE

To determine

To calculate: The solution set for the system of equations,

  $\begin{array}{c}2x+3y=7\\ 3x+4y=10\end{array}$

Answer to Problem 2WE

The solution set of system of equations is $\left(2,1\right)$ .

Explanation of Solution

Given information:

The system of equations are

  $\begin{array}{c}2x+3y=7\\ 3x+4y=10\end{array}$

Formula used:

For system of equations

  $\begin{array}{c}ax+by=e\\ cx+dy=f\end{array}$

The determinant $D=\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|,D_x=\left|\begin{array}{cc}e& b\\ f& d\end{array}\right|\text{ and }{D}_y=\left|\begin{array}{cc}a& e\\ c& f\end{array}\right|$ .

The values of x and y are $x=\frac{D_x}{D}\text{ and }y=\frac{D_y}{D}$.

Calculation:

Consider the system of equations are

  $\begin{array}{c}2x+3y=7\\ 3x+4y=10\end{array}$

Recall that for system of equations

  $\begin{array}{c}ax+by=e\\ cx+dy=f\end{array}$

The determinant $D=\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|,D_x=\left|\begin{array}{cc}e& b\\ f& d\end{array}\right|\text{ and }{D}_y=\left|\begin{array}{cc}a& e\\ c& f\end{array}\right|$ .

The values of x and y are $x=\frac{D_x}{D}\text{ and }y=\frac{D_y}{D}$ .

Apply it,

  $\begin{array}{c}D=\left|\begin{array}{cc}2& 3\\ 3& 4\end{array}\right|\\ =8-9\\ =-1\end{array}$

And,

  $\begin{array}{c}{D}_x=\left|\begin{array}{cc}7& 3\\ 10& 4\end{array}\right|\\ =28-30\\ =-2\end{array}$

And,

  $\begin{array}{c}{D}_y=\left|\begin{array}{cc}2& 7\\ 3& 10\end{array}\right|\\ =20-21\\ =-1\end{array}$

The values of x and y are $x=\frac{D_x}{D}\text{ and }y=\frac{D_y}{D}$ .

Apply it,

  $\begin{array}{c}x=\frac{D_x}{D}\\ =\frac{-2}{-1}\\ =2\end{array}$

And,

  $\begin{array}{c}y=\frac{D_y}{D}\\ =\frac{-1}{-1}\\ =1\end{array}$

Thus, the solution set of system of equations is $\left(2,1\right)$ .