Question

complete the sentence below by replacing ? with an appropriate value. using cramer’s rule, the value of x that satisfies the system of equations \\( \

$$\begin{cases} 2x + 3y = 5 \\\\ x - 4y = -3 \\end{cases}$$

\\) is \\( x = \frac{?}{\

$$\begin{vmatrix} 2 & 3 \\\\ 1 & -4 \\end{vmatrix}$$

} \\). choose the correct answer below. \\( \

$$\begin{vmatrix} 5 & 2 \\\\ -3 & 1 \\end{vmatrix}$$

\\) \\( \

$$\begin{vmatrix} 5 & 3 \\\\ -3 & -4 \\end{vmatrix}$$

\\) \\( \

$$\begin{vmatrix} 3 & 2 \\\\ -4 & 1 \\end{vmatrix}$$

\\) \\( \

$$\begin{vmatrix} 2 & 5 \\\\ 1 & -3 \\end{vmatrix}$$

\\)

Explanation:

Step1: Recall Cramer's Rule for x

For a system \(

$$\begin{cases}a_1x + b_1y = c_1\\a_2x + b_2y = c_2\end{cases}$$

\), \(x=\frac{

$$\begin{vmatrix}c_1&b_1\\c_2&b_2\end{vmatrix}$$

}{

$$\begin{vmatrix}a_1&b_1\\a_2&b_2\end{vmatrix}$$

}\).
Here, \(a_1 = 2\), \(b_1 = 3\), \(c_1 = 5\), \(a_2 = 1\), \(b_2 = -4\), \(c_2 = -3\).

Step2: Compute numerator determinant

The numerator for \(x\) is \(

$$\begin{vmatrix}c_1&b_1\\c_2&b_2\end{vmatrix}$$

=

$$\begin{vmatrix}5&3\\-3&-4\end{vmatrix}$$

\).

Answer:

\(

$$\begin{vmatrix}5&3\\-3&-4\end{vmatrix}$$

\) (corresponding to the second option)