Question

How do you determine whether a linear system has one solution, many solutions, or no solution when given x+ 5y=1 and -3x +4y= 16?

Answer 1

Rearrange in slope intercept form and notice that the slopes are different, so these equations represent two non-parallel lines that intersect at one point - that is one solution.

Explanation:

Given `x+5y=1`, subtract `x` from both sides, then divide both sides by `5` to get:

`y = -1/5 x + 1/5`

Given `-3x+4y=16`, add `3x` to both sides, then divide both sides by `4` to get:

`y = 3/4 x + 4`

So these equations represent lines with slope `-1/5` and `3/4`.

Since the slopes are different, the lines will intersect at exactly one point.

graph{(x+5y-1)(-3x+4y-16) = 0 [-10, 10, -4.5, 5.5]}