Chapter 11.3, Problem 2AYU

To determine

To fill: The blank in the statement “Using Cramer’s Rule, the value of x that satisfies the system of equations $\{\begin{array}{c}2x+3y=5\\ x-4y=-3\end{array}\text{ is }x=\frac{\underset{\_}{}}{\left|\begin{array}{cc}2& 3\\ 1& -4\end{array}\right|}$ ”.

Answer to Problem 2AYU

The complete statement is “Using Cramer’s Rule, the value of x that satisfies the system of equations $\{\begin{array}{c}2x+3y=5\\ x-4y=-3\end{array}\text{ is }x=\frac{\left|\begin{array}{cc}5& 3\\ -3& -4\end{array}\right|}{\left|\begin{array}{cc}2& 3\\ 1& -4\end{array}\right|}$ ”.

Explanation of Solution

Given information:

The statement “Using Cramer’s Rule, the value of x that satisfies the system of equations $\{\begin{array}{c}2x+3y=5\\ x-4y=-3\end{array}\text{ is }x=\frac{\underset{\_}{}}{\left|\begin{array}{cc}2& 3\\ 1& -4\end{array}\right|}$ ”.

Consider the provided statement “Using Cramer’s Rule, the value of x that satisfies the system of equations $\{\begin{array}{c}2x+3y=5\\ x-4y=-3\end{array}\text{ is }x=\frac{\underset{\_}{}}{\left|\begin{array}{cc}2& 3\\ 1& -4\end{array}\right|}$ ”.

Consider the system of equations $\{\begin{array}{c}2x+3y=5\\ x-4y=-3\end{array}$ , the system is solved with help of Cramer’s Rule as first write the system in the form of augmented matrix,

  $\left[\begin{array}{cc}2& 3\\ 1& -4\end{array}|\begin{array}{c}5\\ -3\end{array}\right]$

The value of variables are $x=\frac{{△}_1}{\Delta }$ and $y=\frac{{△}_2}{\Delta }$ .

Here $\Delta$ is the determinant of the matrix and ${△}_1$ is the determinant of the terms excluding the coefficient of y and ${△}_2$ is the determinant of the terms excluding the coefficient of x .

So, $\Delta =\left|\begin{array}{cc}2& 3\\ 1& -4\end{array}\right|,{\Delta }_1=\left|\begin{array}{cc}5& 3\\ -3& -4\end{array}\right|\text{ and }{\Delta }_2=\left|\begin{array}{cc}2& 5\\ 1& -3\end{array}\right|$ .

Therefore, solution is $x=\frac{\left|\begin{array}{cc}5& 3\\ -3& -4\end{array}\right|}{\left|\begin{array}{cc}2& 3\\ 1& -4\end{array}\right|}$ and $y=\frac{\left|\begin{array}{cc}2& 5\\ 1& -3\end{array}\right|}{\left|\begin{array}{cc}2& 3\\ 1& -4\end{array}\right|}$ .

Thus, the complete statement is “Using Cramer’s Rule, the value of x that satisfies the system of equations $\{\begin{array}{c}2x+3y=5\\ x-4y=-3\end{array}\text{ is }x=\frac{\left|\begin{array}{cc}5& 3\\ -3& -4\end{array}\right|}{\left|\begin{array}{cc}2& 3\\ 1& -4\end{array}\right|}$ ”.