Solve Using a Matrix by Elimination x-2y+3x=9 , -x+3y=-4 , 2x-5y+5z=17

Problem

{[x−2*y+3*z=9],[−x+3*y=−4],[2*x−5*y+5*z=17])

Solution

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

[[1,−2,3,9],[−1,3,0,−4],[2,−5,5,17]]

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

[[1,−2,3,9],[0,1,3,5],[0,−1,−1,−1]]

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

[[1,−2,3,9],[0,1,3,5],[0,0,2,4]]

4. Simplify the third row by dividing by 2 ((R_3)/2→(R_3) to solve for z

[[1,−2,3,9],[0,1,3,5],[0,0,1,2]]

⇒z=2

5. Substitute z=2 into the second row equation y+3*z=5 to solve for y

y+3*(2)=5

y+6=5

⇒y=−1

6. Substitute y=−1 and z=2 into the first row equation x−2*y+3*z=9 to solve for x

x−2*(−1)+3*(2)=9

x+2+6=9

x+8=9

⇒x=1

Final Answer

(x,y,z)=(1,−1,2)

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