Question

How do you determine whether a linear system has one solution, many solutions, or no solution when given x + 6y = 28 and 2x - 3y = -19?

Answer 1

Rearrange the two equations into slope intercept format and compare the slopes. Since the slopes are different we have a pair of intersecting lines, that is exactly one solution.

Explanation:

Given `x+6y=28`, subtract `x` from both sides and divide both sides by `6` to get:

`y = -1/6 x + 14/3`

Given `2x-3y = -19`, subtract `2x` from both sides and divide both sides by `-3` to get:

`y = 2/3 x + 19/3`

Both of these equations are now in slope intercept form, so we can see that the slopes of the two lines represented by the equations are different, the first being `-1/6` and the second `2/3`.

So the two lines are not parallel and they intersect at exactly one point. That is the two equations can be satisfied simultaneously for exactly one pair of `x, y` values.

graph{(x+6y-28)(2x-3y+19) = 0 [-10.625, 9.375, -2.2, 7.8]}