Solve Using a Matrix by Elimination

Problem

{[x−7*y+7*z=−27],[−6*x+y+8*z=−38],[2*x+3*y+7*z=−26])

Solution

1. Write the augmented matrix representing the system of linear equations.

[[1,−7,7,|,−27],[−6,1,8,|,−38],[2,3,7,|,−26]]

2. Eliminate the x-terms in the second and third rows by performing row operations (R_2)←(R_2)+6*(R_1) and (R_3)←(R_3)−2*(R_1)

[[1,−7,7,|,−27],[0,−41,50,|,−200],[0,17,−7,|,28]]

3. Eliminate the y-term in the third row by performing the operation (R_3)←41*(R_3)+17*(R_2)

41*(17)+17*(−41)=0

41*(−7)+17*(50)=−287+850=563

41*(28)+17*(−200)=1148−3400=−2252

[[1,−7,7,|,−27],[0,−41,50,|,−200],[0,0,563,|,−2252]]

4. Solve for z using the third row.

563*z=−2252

z=−4

5. Solve for y using back-substitution into the second row.

−41*y+50*(−4)=−200

−41*y−200=−200

−41*y=0

y=0

6. Solve for x using back-substitution into the first row.

x−7*(0)+7*(−4)=−27

x−28=−27

x=1

Final Answer

(x,y,z)=(1,0,−4)

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