Solve by using matrix method. 4x 3y = 5 6x + 2y = 40

To solve the given system of linear equations using the matrix method (also known as Cramer’s rule), we first express the system in matrix form $AX=B$, where:

$$A=\left[\begin{array}{cc}4& -3\\ 6& 2\end{array}\right],X=\left[\begin{array}{c}x\\ y\end{array}\right],B=\left[\begin{array}{c}5\\ 40\end{array}\right]$$

Step 1: Compute the determinant of $A$ ($|A|$)

$$|A|=\left|\begin{array}{cc}4& -3\\ 6& 2\end{array}\right|=(4)(2)-(-3)(6)=8+18=26$$

Since $|A|\ne 0$, the system has a unique solution.

Step 2: Compute $x$ using Cramer’s rule

Replace the first column of $A$ with $B$ to form $A_x$:

A_x = \begin{bmatrix} 5 & -3 \\ 40 & 2 \end{vmatrix}

$$|A_x|=(5)(2)-(-3)(40)=10+120=130$$

$$x=\frac{|A_x|}{|A|}=\frac{130}{26}=5$$

Step 3: Compute $y$ using Cramer’s rule

Replace the second column of $A$ with $B$ to form $A_y$:

A_y = \begin{bmatrix} 4 & 5 \\ 6 & 40 \end{vmatrix}

$$|A_y|=(4)(40)-(5)(6)=160-30=130$$

$$y=\frac{|A_y|}{|A|}=\frac{130}{26}=5$$

Final Solution

The solution to the system is: $$\boxed{x=5,y=5}$$