Find the Variables [[x+y,2z+w],[y-x,z-w]]=[[3,5],[1,4]]

Problem

[[x+y,2*z+w],[y−x,z−w]]=[[3,5],[1,4]]

Solution

1. Equate the corresponding entries of the two matrices to create a system of linear equations.

x+y=3

y−x=1

2*z+w=5

z−w=4

2. Solve the first system for x and y by adding the first two equations together.

(x+y)+(y−x)=3+1

2*y=4

y=2

3. Substitute the value of y back into the first equation to find x

x+2=3

x=1

4. Solve the second system for z and w by adding the third and fourth equations together.

(2*z+w)+(z−w)=5+4

3*z=9

z=3

5. Substitute the value of z back into the fourth equation to find w

3−w=4

−w=1

w=−1

Final Answer

x=1,y=2,z=3,w=−1

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