Word Problems with Two Lines Examples

A brother and sister each grab a few handfuls of candy. The brother takes 17 hours to eat 51 pieces of candy, while the sister takes 9 hours to eat 27 pieces of candy. No, it's not Halloween. Just a normal day in the Howell household. After how many hours did the two siblings have the same amount of candy left?

The independent variable x is hours, and the dependent variable y is pieces of candy left. Since the brother took 17 hours to eat 51 pieces of candy, he ate 51 ÷ 17 = 3 pieces of candy per hour. If we turn his candy-eating into an equation, we get y = 51 – 3x.

If we turn that into slope-intercept form, we get y = -3x + 51. The slope of this line is -3.

The sister took 9 hours to eat 27 pieces of candy, so she was wolfing it down at a rate of 27 ÷ 9 = 3 pieces per hour. Her equation, then, is y = 27 – 3x, or y = -3x + 27 in slope-intercept form. That means we're trying to solve this system of equations:

y = -3x + 51

y = -3x + 27

Each line has a slope of -3. Since the lines are parallel, the system of linear equations they represent is inconsistent. The lines will never intersect, so the brother and sister will never have the same amount of candy. Until, of course, they're both left completely candy-less, at which point they simultaneously bust out into a good cry.