2310.31 – Intersecting Lines

For which real values of mm 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−1m = 2m - 1, or m=1m = 1. The answer is (d).