Question

solve the system by cramer’s rule.
$x + 2y = 11$
$2x + y = 7$
the solution to the system is
(simplify your answer. type an ordered pair.)

Explanation:

Step1: Identify the system

The system is \(

$$\begin{cases} x + 2y = 11 \\ 2x + y = 7 \end{cases}$$

\). For Cramer's rule, first find the coefficient matrix \( A =

$$\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$$

\).

Step2: Calculate determinant of A (D)

The determinant \( D = (1)(1) - (2)(2) = 1 - 4 = -3 \).

Step3: Calculate \( D_x \) (replace x - column with constants)

\( D_x =

$$\begin{vmatrix} 11 & 2 \\ 7 & 1 \end{vmatrix}$$

= (11)(1) - (2)(7) = 11 - 14 = -3 \).

Step4: Calculate \( D_y \) (replace y - column with constants)

\( D_y =

$$\begin{vmatrix} 1 & 11 \\ 2 & 7 \end{vmatrix}$$

= (1)(7) - (11)(2) = 7 - 22 = -15 \).

Step5: Find x and y

\( x = \frac{D_x}{D} = \frac{-3}{-3} = 1 \), \( y = \frac{D_y}{D} = \frac{-15}{-3} = 5 \).

Answer:

\((1, 5)\)