Question

How do you determine whether a linear system has one solution, many solutions, or no solution when given 4x-6y=2 and 5x+3y= 1?

Answer 1

Compare the slopes of the two linear equations.
Since the slopes are not equal there is 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).

`4x-6y=2` has a slope of `-4/(-6) = 2/3`

`5x+3y=1` has a slope of `-5/3`

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