Find the Variables

Problem

[[0,−1,1],[2,2,−5],[−1,−1,3]]*[[x],[y],[z]]=[[8],[15],[23]]

Solution

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

−y+z=8

2*x+2*y−5*z=15

−x−y+3*z=23

2. Isolate x in the third equation to express it in terms of z and y

−x=−y+3*z−23

x=y−3*z+23

3. Substitute the expression for x into the second equation.

2*(y−3*z+23)+2*y−5*z=15

2*y−6*z+46+2*y−5*z=15

4*y−11*z=−31

4. Solve the first equation for y in terms of z

−y=8−z

y=z−8

5. Substitute the expression for y into the simplified second equation to find z

4*(z−8)−11*z=−31

4*z−32−11*z=−31

−7*z=1

z=−1/7

6. Calculate y using the value of z

y=−1/7−8

y=−1/7−56/7

y=−57/7

7. Calculate x using the values of y and z

x=−57/7−3*(−1/7)+23

x=−57/7+3/7+161/7

x=107/7

Final Answer

[[x],[y],[z]]=[[107/7],[−57/7],[−1/7]]

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