Find the Variables

Problem

4*[[x,y],[z,−1]]=2*[[y,z],[−x,1]]+2*[[3,x],[9,−x]]

Solution

1. Distribute the scalar constants into each matrix by multiplying every element inside the brackets.

4*[[x,y],[z,−1]]=[[4*x,4*y],[4*z,−4]]

2*[[y,z],[−x,1]]=[[2*y,2*z],[−2*x,2]]

2*[[3,x],[9,−x]]=[[6,2*x],[18,−2*x]]

2. Add the two matrices on the right side of the equation by summing their corresponding entries.

[[4*x,4*y],[4*z,−4]]=[[2*y+6,2*z+2*x],[−2*x+18,2−2*x]]

3. Set up a system of linear equations by equating the corresponding elements from the left and right matrices.

4*x=2*y+6

4*y=2*z+2*x

4*z=−2*x+18

−4=2−2*x

4. Solve for x using the fourth equation.

−4=2−2*x

−6=−2*x

x=3

5. Substitute x=3 into the third equation to solve for z

4*z=−2*(3)+18

4*z=12

z=3

6. Substitute x=3 and z=3 into the second equation to solve for y

4*y=2*(3)+2*(3)

4*y=12

y=3

7. Verify the values using the first equation.

4*(3)=2*(3)+6

12=12

Final Answer

x=3,y=3,z=3

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