Find the Variables

Problem

[[a+8,5*z+1,5*m],[5*k,4,4]]+[[6*a,8*z,8*m],[6*k,5,5]]=[[36,−12,78],[11,9,9]]

Solution

1. Add the matrices on the left side by summing the corresponding entries.

[[(a+8)+6*a,(5*z+1)+8*z,5*m+8*m],[5*k+6*k,4+5,4+5]]=[[36,−12,78],[11,9,9]]

2. Simplify the expressions within the resulting matrix.

[[7*a+8,13*z+1,13*m],[11*k,9,9]]=[[36,−12,78],[11,9,9]]

3. Set up equations by equating corresponding entries from both sides of the matrix equation.

7*a+8=36

13*z+1=−12

13*m=78

11*k=11

4. Solve for a by subtracting 8 and dividing by 7.

7*a=28

a=4

5. Solve for z by subtracting 1 and dividing by 13.

13*z=−13

z=−1

6. Solve for m by dividing by 13.

m=78/13

m=6

7. Solve for k by dividing by 11.

k=1

Final Answer

a=4,z=−1,m=6,k=1

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