Simplify the Matrix

Problem

[[−3*x,2*y,4*z],[x,6*y,−2*z],[4*x,4*y,−6*z]]=[[−22],[16],[38]]

Solution

1. Interpret the matrix equation as a system of linear equations by equating the corresponding rows.

2. Write the system of equations:

−3*x+2*y+4*z=−22

x+6*y−2*z=16

4*x+4*y−6*z=38

3. Eliminate z by adding twice the second equation to the first equation:

(−3*x+2*y+4*z)+2*(x+6*y−2*z)=−22+2*(16)

−x+14*y=10

4. Eliminate z again by subtracting three times the second equation from the third equation:

(4*x+4*y−6*z)−3*(x+6*y−2*z)=38−3*(16)

x−14*y=−10

5. Observe that the two resulting equations are dependent (multiples of each other), meaning there are infinitely many solutions.

6. Solve for x in terms of y using the equation −x+14*y=10

x=14*y−10

7. Substitute x back into the second original equation to solve for z in terms of y

(14*y−10)+6*y−2*z=16

20*y−10−2*z=16

−2*z=−20*y+26

z=10*y−13

8. Express the solution set using y as a free parameter t

Final Answer

[[x],[y],[z]]=[[14*t−10],[t],[10*t−13]]

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