2310.31 – Intersecting Lines

For which real values of m are the simultaneous equations

y=mx+3 and y=(2m–1)x+4y = mx + 3 \text{ and }y = (2m – 1)x + 4

satisfied by at least one pair of real numbers (x,y)(x, y)?

  1. all m>0m > 0
  2. all m≠0m \neq 0
  3. all m≠.5m \neq .5
  4. all m≠1m \neq 1
  5. all values of mm
Solution

The simultaneous equations will have a real solution unless the lines are parallel. The lines are parallel if their slopes are the same, that is, when m = 2m - 1, or m = 1. The answer is (d).