Use Matrix method to solve the following x+y=5 and 3x−2y=0

$$\begin{bmatrix} 1 & 1 \\ 3 & -2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ 0 \end{bmatrix}$$

Hapa, $A = \begin{bmatrix} 1 & 1 \\ 3 & -2 \end{bmatrix}$, $X = \begin{bmatrix} x \\ y \end{bmatrix}$, na $B = \begin{bmatrix} 5 \\ 0 \end{bmatrix}$.

Kwanza, tafuta determinant ya $A$:

$$\det(A) = (1)(-2) - (1)(3) = -2 - 3 = -5$$

Kigeuzi cha $A$ ni:

$$A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} = \frac{1}{-5} \begin{bmatrix} -2 & -1 \\ -3 & 1 \end{bmatrix}$$ $$A^{-1} = \begin{bmatrix} \frac{-2}{-5} & \frac{-1}{-5} \\ \frac{-3}{-5} & \frac{1}{-5} \end{bmatrix} = \begin{bmatrix} \frac{2}{5} & \frac{1}{5} \\ \frac{3}{5} & -\frac{1}{5} \end{bmatrix}$$

$$\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} \frac{2}{5} & \frac{1}{5} \\ \frac{3}{5} & -\frac{1}{5} \end{bmatrix} \begin{bmatrix} 5 \\ 0 \end{bmatrix}$$

Fanya kuzidisha kwa matriki:

$$x = \left(\frac{2}{5}\right)(5) + \left(\frac{1}{5}\right)(0) = 2 + 0 = 2$$ $$y = \left(\frac{3}{5}\right)(5) + \left(-\frac{1}{5}\right)(0) = 3 + 0 = 3$$

Hivyo, suluhisho ni:

$$x = 2, y = 3$$