Solve Using a Matrix by Elimination

Problem

{[−8*x+6*y−4*z=2],[−2*x−6*y+7*z=24],[−6*x+12*y+5*z=10])

Solution

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

[[−8,6,−4,2],[−2,−6,7,24],[−6,12,5,10]]

2. Simplify the first row by dividing by −2 to create a leading value of 4 (or simplify the arithmetic).

(R_1)→(R_1)/(−2)

[[4,−3,2,−1],[−2,−6,7,24],[−6,12,5,10]]

3. Eliminate the x terms in the second and third rows using the first row.

(R_2)→(R_2)+1/2*(R_1)

(R_3)→(R_3)+3/2*(R_1)

[[4,−3,2,−1],[0,−7.5,8,23.5],[0,7.5,8,8.5]]

4. Eliminate the y term in the third row by adding the second row to the third row.

(R_3)→(R_3)+(R_2)

[[4,−3,2,−1],[0,−7.5,8,23.5],[0,0,16,32]]

5. Solve for z using the third row.

16*z=32

z=2

6. Substitute z=2 into the second row to solve for y

−7.5*y+8*(2)=23.5

−7.5*y+16=23.5

−7.5*y=7.5

y=−1

7. Substitute y=−1 and z=2 into the first row to solve for x

4*x−3*(−1)+2*(2)=−1

4*x+3+4=−1

4*x+7=−1

4*x=−8

x=−2

Final Answer

(x,y,z)=(−2,−1,2)

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