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For a certain value of k, the system

x + y + 3z = 10,
-4x + 3y + 5z = 7,
kx + z = 3
has no solutions. What is this value of k?

Sagot :

mhanifa

Answer:

  • k = 7/4

Step-by-step explanation:

Given system:

  • x + y + 3z = 10,
  • -4x + 3y + 5z = 7,
  • kx + z = 3

Substitute z in the first two equations with z = 3 - kx:

  • x + y + 3(3 - kx) = 10 ⇒ (1 - 3k)x + y = 1 ⇒ y = (3k - 1)x + 1
  • -4x + 3y +5(3 - kx) = 7 ⇒ -(4 + 5k)x + 3y = -8 ⇒ y = 1/3(5k + 4) - 8

In order the two equations above not to have a solution, the lines should be parallel. Parallel lines have equal slopes.

Solve the following equation for k:

  • 3k - 1 = 1/3(5k + 4)
  • 9k - 3 = 5k + 4
  • 4k = 7
  • k = 7/4
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