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# Solve for x and y $\begin{bmatrix} 3 & \quad -4 \\ 1 & \quad 2 \end{bmatrix}$ $\begin{bmatrix} x \\ y \end{bmatrix}$ = $\begin{bmatrix} 3 \\ 11 \end{bmatrix}$

$\begin{array}{1 1} x = 15,y = 3 \\ x = 5,y = 3 \\ x = -5,y = -3 \\ x = 5,y = 0 \end{array}$

Toolbox:
• If the order of 2 matrices are equal, their corresponding elements are equal, i.e, if $A_{ij}=B_{ij}$, then any element $a_{ij}$ in matrix A is equal to corresponding element $b_{ij}$ in matrix B.
• We can then match the corresponding elements and solve the resulting equations to find the values of the unknown variables.
• If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B: $\begin{bmatrix}AB\end{bmatrix}_{i,j} = A_{i,1}B_{1,j} + A_{i,2}B_{2,j} + A_{i,3}B_{3,j} ... A_{i,n}B_{n,j}$
Step1:
Given
$\begin{bmatrix} 3 & \quad -4 \\ 1 & \quad 2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 3 \\ 11 \end{bmatrix}$
$\begin{bmatrix}3(x)-4y & x+2y\end{bmatrix}=\begin{bmatrix}3\\11\end{bmatrix}$
$\Rightarrow \begin{bmatrix}3x-4y\\x+2y\end{bmatrix}=\begin{bmatrix}3\\11\end{bmatrix}$
The above two matrices are equal,hence their corresponding elements should be equal.
Step2:
3x-4y=3-----(1)
x+2y=11-----(2)
Multiply equation (2) by 2 and add with (1)
3x-4y=3
2x+4y=22
________________
5x=25
x=$\frac{25}{5}$=5
x=5
Step3:
Substitute the value of x in equation (2)
5+2y=11
2y=11-5
2y=6
y=3.
Therefore x=5,y=3.