Find the Variables

Problem

[[1,−2,−3],[−1,4,6],[1,−1,−2]]*[[q],[x],[b]]=[[7],[7],[2]]

Solution

1. Write the augmented matrix representing the system of linear equations to prepare for Gaussian elimination.

[[1,−2,−3,|,7],[−1,4,6,|,7],[1,−1,−2,|,2]]

2. Eliminate the first column entries below the pivot by adding the first row to the second row ((R_2)+(R_1)⇒(R_2) and subtracting the first row from the third row ((R_3)−(R_1)⇒(R_3).

[[1,−2,−3,|,7],[0,2,3,|,14],[0,1,1,|,−5]]

3. Swap rows to make the arithmetic easier, moving the row with the smaller coefficient to the second position ((R_2)↔(R_3).

[[1,−2,−3,|,7],[0,1,1,|,−5],[0,2,3,|,14]]

4. Eliminate the second column entry below the pivot by subtracting twice the second row from the third row ((R_3)−2*(R_2)⇒(R_3).

[[1,−2,−3,|,7],[0,1,1,|,−5],[0,0,1,|,24]]

5. Solve for b using the third row of the resulting upper triangular matrix.

b=24

6. Solve for x by substituting b=24 into the second row equation x+b=−5

x+24=−5

x=−29

7. Solve for q by substituting x=−29 and b=24 into the first row equation q−2*x−3*b=7

q−2*(−29)−3*(24)=7

q+58−72=7

q−14=7

q=21

Final Answer

[[q],[x],[b]]=[[21],[−29],[24]]

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