Solve Using a Matrix by Elimination 3x+2y+6z=5 , 6x+3y+7z=8 , 4x+5y-7z=10

Problem

{[3*x+2*y+6*z=5],[6*x+3*y+7*z=8],[4*x+5*y−7*z=10])

Solution

1. Write the system of equations as an augmented matrix.

[[3,2,6,|,5],[6,3,7,|,8],[4,5,−7,|,10]]

2. Eliminate the x terms in the second and third rows by performing row operations (R_2)−2*(R_1)→(R_2) and (R_3)−4/3*(R_1)→(R_3)

[[3,2,6,|,5],[0,−1,−5,|,−2],[0,7/3,−15,|,10/3]]

3. Eliminate the y term in the third row by performing the operation (R_3)+7/3*(R_2)→(R_3)

[[3,2,6,|,5],[0,−1,−5,|,−2],[0,0,−80/3,|,−4/3]]

4. Solve for z using the third row.

−80/3*z=−4/3

z=4/80

z=1/20

5. Substitute z into the second row equation to solve for y

−y−5*(1/20)=−2

−y−1/4=−2

−y=−7/4

y=7/4

6. Substitute y and z into the first row equation to solve for x

3*x+2*(7/4)+6*(1/20)=5

3*x+7/2+3/10=5

3*x+35/10+3/10=5

3*x+38/10=5

3*x=5−19/5

3*x=6/5

x=2/5

Final Answer

(x,y,z)=(2/5,7/4,1/20)

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