Find the Variables

Problem

[[−3,1,2],[4,2,−4],[0,1,3]]*[[x],[y],[z]]=[[5],[−4],[3]]

Solution

1. Write the system of linear equations by performing matrix multiplication on the left side.

−3*x+y+2*z=5

4*x+2*y−4*z=−4

y+3*z=3

2. Simplify the second equation by dividing all terms by 2

2*x+y−2*z=−2

3. Eliminate x by adding the first equation to 1.5 times the simplified second equation, or more simply, multiply the first by 2 and the second by 3 and add them.

2*(−3*x+y+2*z)=2*(5)⇒−6*x+2*y+4*z=10

3*(2*x+y−2*z)=3*(−2)⇒6*x+3*y−6*z=−6

(−6*x+2*y+4*z)+(6*x+3*y−6*z)=10−6⇒5*y−2*z=4

4. Solve the system of two equations involving y and z using the result from step 3 and the third original equation.

5*y−2*z=4

y+3*z=3⇒y=3−3*z

5. Substitute y=3−3*z into the equation 5*y−2*z=4

5*(3−3*z)−2*z=4

15−15*z−2*z=4

−17*z=−11

z=11/17

6. Find y by substituting z back into the expression for y

y=3−3*(11/17)

y=51/17−33/17

y=18/17

7. Find x by substituting y and z into the first original equation.

−3*x+18/17+2*(11/17)=5

−3*x+18/17+22/17=5

−3*x+40/17=85/17

−3*x=45/17

x=−15/17

Final Answer

[[x],[y],[z]]=[[−15/17],[18/17],[11/17]]

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