Determine if Linear

Problem

[[2,−4,6],[1,−3,5],[3,−7,12]]*[[(x_1)],[(x_2)],[(x_3)]]=[[14],[7],[2]]

Solution

1. Set up the augmented matrix representing the system of linear equations.

[[2,−4,6,|,14],[1,−3,5,|,7],[3,−7,12,|,2]]

2. Swap the first and second rows to create a leading 1 in the first row.

[[1,−3,5,|,7],[2,−4,6,|,14],[3,−7,12,|,2]]

3. Eliminate the entries below the first pivot by performing (R_2)−2*(R_1)→(R_2) and (R_3)−3*(R_1)→(R_3)

[[1,−3,5,|,7],[0,2,−4,|,0],[0,2,−3,|,−19]]

4. Eliminate the entry below the second pivot by performing (R_3)−(R_2)→(R_3)

[[1,−3,5,|,7],[0,2,−4,|,0],[0,0,1,|,−19]]

5. Identify the rank of the coefficient matrix and the augmented matrix. Since there are three pivots for three variables, the system is consistent and has a unique solution.

6. Solve for the variables using back-substitution starting with (x_3)=−19

2*(x_2)−4*(−19)=0⇒2*(x_2)+76=0⇒(x_2)=−38

(x_1)−3*(−38)+5*(−19)=7⇒(x_1)+114−95=7⇒(x_1)+19=7⇒(x_1)=−12

Final Answer

[[(x_1)],[(x_2)],[(x_3)]]=[[−12],[−38],[−19]]

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