Question

Find k if the equations 2x + 3y – z = 5, 3x – y + 4z = 2, x + 7y – 6z = k are consistent

Answer 1

2x + 3y – z = 5

3x – y + 4z = 2

x + 7y – 6z = k

The matrix form of these equations

`[(2, 3, -1),(3, -1, 4),(1, 7, -6)] [(x),(y),(z)] = [(5),(2),("k")]`
          A                X    =     B

Augmented Matrix [A, B]	Elementary Transformation
`[(2, 3, -1, 5),(3, -1, 4, 2),(1, 7, -6, "k")]`
`[(1, 7, -6, "k"),(3, -1, 4, 2),(2, 3, -1, 5)]`	`{:"R"_1 ↔ "R"_3:}`
`[(1, 7, -6, "k"),(0, -22, 22, 2 - 3"k"),(0, -11, 11,  5 - 2"k")]`	`{:("R"_2 -> "R"_2 - 3"R"_1),("R"_3 -> "R"_3 - 2"R"_1):}`
`[(1, 7, -6, "k"),(0, -11, 11,  5 - 2"k"),(0, -22, 22, 2 - 3"k")]`	`{:"R"_2 ↔ "R"_3:}`
`[(1, 7, -6, "k"),(0, -11, 11, 5 - 2"k"),(0, 0, 0, "k" - 8)]`	`{:"R"_3 -> "R"_3 - 2"R"_2:}`
p(A) = 2; p(A, B) = 2 if k = 8

Obviously, the last equivalent matrix is in the ech-elon form.

Since the equations are consistent

p(– A) = p(A, B)

p(A) = 2 and p(A, B) = 2 then

k = 8