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?

1 Answer
Oct 25, 2015

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]}