For example, Maribel has a lemonade stand. She's only 7 years old, so she doesn't want to stand around and be bored. She wants at least three people to come by every 10 minutes. If it's 65 degrees outside, only one person will come by every 10 minutes, while if it's 80 degrees, she'll get six people in 10 minutes. Assuming that's a linear relationship, what temperature should Maribel wait for before starting her sales if she wants at least three people every 10 minutes? You could solve this with equations, but you also can use a graph. Where x is 65, y is 1; where x is 80, y is 6. Connect those two points in a line. What is the x-value where y equals 3?
Steve needs to move a pile of bricks from one end of the construction site to the other. He wants to make as few trips as possible, so he loads the bricks into a wheelbarrow. The tire on the wheelbarrow will go flat if he loads more than 167 pounds. How many 6-pound bricks can he load? Again, it can be solved algebraically, but it also can be graphed. Because zero bricks weigh zero pounds, the first point on the graph can be x = 0 and y = 0. Forty bricks weigh 40 * 6 = 240 pounds, so the next point on the graph can be at y = 240 and x = 40. Connect the two points. What is the x-value where y equals 167? How many whole bricks can he carry?
The Milltown Library has a collection of 1,700 books. They get 120 new books every year. Angela reads one library book every day, except for Feb. 29 on leap years. She's 10 years old, and she'll be heading to college when she turns 18. Will she run out of books to read before she leaves town? Now there are two lines to graph. On day zero, the library has 1,700 books -- let x = 0 and y = 1700. Ten years later, on day 3,650 (ignoring leap days, since Angela skips those anyway), the library will have 1,200 more books, or a total of 2,900 books. So the next point goes at y = 2900 and x = 3650. Connect those points to represent the number of books at any time. The other line, representing the number of books Angela has read, starts at 0 on day 0, and 10 years later, y = 3650 and x = 3650. Connect those two points. Do the lines intersect? Do they intersect before Angela is 18 years old?
Linear equations represent situations where an output always changes the same amount for the same change in input. This applies to an incredible variety of situations --- two billiard balls weigh twice as much as one, two umbrellas block twice as much rain as one, half a potato has half as many calories as a whole potato. The kinds of problems that can be solved with linear equations, and by graphing linear equations, are infinite. The limit is only your imagination.