Question

How many solutions does the following system have:

4x + 8y = 16
3x + 6y = 9
Explain how to determine the number of solutions without solving the system. Then apply elimination, and interpret the resulting expression.

Answer 1

Put them both into slope intercept form and examine the slopes.

If the slopes are different, there is only one solution.

If the slopes are the same, examine the vertical intercepts.

If the intercepts are the same it has infinitely many solutions, if the intercepts are different it has no solutions.

4x + 8y = 16 ⇔ x + 2y = 4
3x + 6y = 9 ⇔ x + 2y = 3

Equations contradict each other (4 = 3).
Therefore, system has no solution.