Question - Solving a Matrix Equation for x and y
Solution:
You want to solve the matrix equation to find the values of x and y. The equation is given as:\[ \begin{pmatrix} 4 & 2 \\ -2 & x \end{pmatrix} \times \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} y & 10 \\ -2 & 4 \end{pmatrix} \]To find the values of x and y, you need to multiply the first two matrices and set the resulting matrix equal to the one with y and x:First, compute the product of the two matrices on the left side:\[ \begin{pmatrix} 4 & 2 \\ -2 & x \end{pmatrix} \times \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 4(1) + 2(0) & 4(2) + 2(1) \\ -2(1) + x(0) & -2(2) + x(1) \end{pmatrix} \]\[ = \begin{pmatrix} 4 & 8 + 2 \\ -2 & -4 + x \end{pmatrix} \]\[ = \begin{pmatrix} 4 & 10 \\ -2 & x - 4 \end{pmatrix} \]Now, equate this resulting matrix to the matrix on the right side of the equation and solve for x and y:\[ \begin{pmatrix} 4 & 10 \\ -2 & x - 4 \end{pmatrix} = \begin{pmatrix} y & 10 \\ -2 & 4 \end{pmatrix} \]For the matrices to be equal, their corresponding elements must be equal, so:\[ y = 4 \quad \text{and} \quad x - 4 = 4 \]From the second equation:\[ x = 4 + 4 \]\[ x = 8 \]Therefore, the values of x and y are 8 and 4, respectively.
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