Question

Find the set of equations having no solution.
x + y - 4 = 0
x + 2y = 2
5x + 2y -12 = 0
10x + 4y - 24 = 0
y = 4x + 7
y = 3x + 5
y + x = 10
y + x = 5

Answer 1

Interpreting the equations as four pairs:
- Pair A: x + y − 4 = 0 and x + 2y = 2
Solve: subtract first from second: y = −2, then x = 6. (Unique solution.)

- Pair B: 5x + 2y − 12 = 0 and 10x + 4y − 24 = 0
The second is 2× the first, so they are dependent (infinitely many solutions: all (x,y) satisfying 5x+2y=12).

- Pair C: y = 4x + 7 and y = 3x + 5
Solve: 4x+7 = 3x+5 ⇒ x = −2, y = −1. (Unique solution.)

- Pair D: x + y = 10 and x + y = 5
Same left side equals different constants ⇒ inconsistent. No solution.

Therefore the set having no solution is {x + y = 10, x + y = 5}.