Solve Using a Matrix by Elimination x-y+z=3 , x+2y+3z=17 , 2x+3y-4z=-8

Problem

{[x−y+z=3],[x+2*y+3*z=17],[2*x+3*y−4*z=−8])

Solution

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

[[1,−1,1,|,3],[1,2,3,|,17],[2,3,−4,|,−8]]

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

[[1,−1,1,|,3],[0,3,2,|,14],[0,5,−6,|,−14]]

3. Eliminate the y-term from the third row by performing the operation 3*(R_3)−5*(R_2)→(R_3)

[[1,−1,1,|,3],[0,3,2,|,14],[0,0,−28,|,−112]]

4. Solve for z using the third row equation −28*z=−112

z=(−112)/(−28)

z=4

5. Solve for y by substituting z=4 into the second row equation 3*y+2*z=14

3*y+2*(4)=14

3*y+8=14

3*y=6

y=2

6. Solve for x by substituting y=2 and z=4 into the first row equation x−y+z=3

x−2+4=3

x+2=3

x=1

Final Answer

(x,y,z)=(1,2,4)

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