Solving a Matrix Equation for x and y
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.
