Question

How do you determine whether a linear system has one solution, many solutions, or no solution when given 2x - 9y = 1 and 7x - 12y = 23?

Answer 1

Compare the slopes of the given linear equations.
Since they are not equal, there will be exactly one solution.

Explanation:

Linear equations with either many solutions or with no solutions must have the same slope (colinear for many solutions; parallel but displaced for no solutions).

`2x-9y=1` has a slope of `-2/(-9) = 2/9`

`7x-12y=23` has a slope of `-7/(-12) = 7/12`

Since the slopes are different the equations are neither colinear nor parallel.