Find the Variables

Problem

[[2,x],[−1,2]]*[[y,0],[2,1]]=[[8,x],[6,2]]

Solution

1. Perform matrix multiplication on the left side by multiplying rows of the first matrix by columns of the second matrix.

[[(2)*(y)+(x)*(2),(2)*(0)+(x)*(1)],[(−1)*(y)+(2)*(2),(−1)*(0)+(2)*(1)]]=[[8,x],[6,2]]

2. Simplify the entries of the resulting matrix on the left side.

[[2*y+2*x,x],[−y+4,2]]=[[8,x],[6,2]]

3. Equate corresponding entries to create a system of equations for the variables x and y

2*y+2*x=8

−y+4=6

4. Solve for y using the equation derived from the second row, first column.

−y=6−4

−y=2

y=−2

5. Substitute the value of y into the first equation to solve for x

2*(−2)+2*x=8

−4+2*x=8

2*x=12

x=6

Final Answer

[[2,6],[−1,2]]*[[−2,0],[2,1]]=[[8,6],[6,2]]⇒x=6,y=−2

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