Solve by using matrix method. 3x + 5y = 25 2x + 3y = 16

To solve the given system of linear equations using the matrix method (inverse method), we follow these steps:

Given Equations:

$$\{\begin{array}{l}3x+5y=25\text{(1)}\\ 2x+3y=16\text{(2)}\end{array}$$

Step 1: Write the system in matrix form $AX=B$

The system can be expressed as: $$\left[\begin{array}{cc}3& 5\\ 2& 3\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}25\\ 16\end{array}\right]$$ Here,

• $A=\left[\begin{array}{cc}3& 5\\ 2& 3\end{array}\right]$,
• $X=\left[\begin{array}{c}x\\ y\end{array}\right]$,
• $B=\left[\begin{array}{c}25\\ 16\end{array}\right]$.

Step 2: Find the determinant of $A$

$$|A|=\left|\begin{array}{cc}3& 5\\ 2& 3\end{array}\right|=(3)(3)-(5)(2)=9-10=-1$$ Since $|A|\ne 0$, the inverse exists.

Step 3: Find the inverse of $A$

The inverse of a $2\times 2$ matrix $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ is: $$A^{-1}=\frac{1}{|A|}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$$ Applying this: $$A^{-1}=\frac{1}{-1}\left[\begin{array}{cc}3& -5\\ -2& 3\end{array}\right]=\left[\begin{array}{cc}-3& 5\\ 2& -3\end{array}\right]$$

Step 4: Multiply $A^{-1}$ with $B$ to find $X$

$$X=A^{-1}B=\left[\begin{array}{cc}-3& 5\\ 2& -3\end{array}\right]\left[\begin{array}{c}25\\ 16\end{array}\right]$$ Perform the matrix multiplication: $$x=(-3)(25)+(5)(16)=-75+80=5$$ $$y=(2)(25)+(-3)(16)=50-48=2$$

Final Solution:

$$\boxed{x=5,y=2}$$