Question

use cramer’s rule to solve eac

1. $4x - 5y = 39$

$3x + 8y = -6$

1. $-8a - 5b = -27$

Explanation:

Step1: Recall Cramer's Rule for 2x2 system

For system \(

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

\), determinant \( D =

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

= a_1b_2 - a_2b_1 \), \( D_x =

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

= c_1b_2 - c_2b_1 \), \( D_y =

$$\begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix}$$

= a_1c_2 - a_2c_1 \). Then \( x = \frac{D_x}{D} \), \( y = \frac{D_y}{D} \) (if \( D
eq 0 \)).

Step2: Identify coefficients for problem 13

System: \(

$$\begin{cases} 4x - 5y = 39 \\ 3x + 8y = -6 \end{cases}$$

\). So \( a_1 = 4, b_1 = -5, c_1 = 39 \); \( a_2 = 3, b_2 = 8, c_2 = -6 \).

Step3: Calculate D

\( D =

$$\begin{vmatrix} 4 & -5 \\ 3 & 8 \end{vmatrix}$$

= 4(8) - 3(-5) = 32 + 15 = 47 \).

Step4: Calculate \( D_x \)

\( D_x =

$$\begin{vmatrix} 39 & -5 \\ -6 & 8 \end{vmatrix}$$

= 39(8) - (-6)(-5) = 312 - 30 = 282 \).

Step5: Calculate \( D_y \)

\( D_y =

$$\begin{vmatrix} 4 & 39 \\ 3 & -6 \end{vmatrix}$$

= 4(-6) - 3(39) = -24 - 117 = -141 \).

Step6: Find x and y

\( x = \frac{D_x}{D} = \frac{282}{47} = 6 \).
\( y = \frac{D_y}{D} = \frac{-141}{47} = -3 \).

Answer:

For the system \( 4x - 5y = 39 \) and \( 3x + 8y = -6 \), the solution is \( x = 6 \), \( y = -3 \).