XII Using Lines to Solve Problems

These techniques can be used to solve problems that involve relationships between sets of values. Assuming we know that 40 pieces of some article cost $70 and that 75 pieces of the same article cost $131.25, their relationships can be used to figure out the individual cost of the article. If we imagine each quantity and price pair to be a point we can construct a line showing the relationship between the two by putting both quantities and both prices on a shared axis each. The slope in this example would reveal the individual price for a single article.

Substitution can be used to solve problems involving relationships as well. Example:

A boy and a girl have a total of 57 friends. The girl has twice as many friends as the boy. How many do each have? This problem can be set up and solved with the following equations that follow from the description:

a+b=57a + b = 57, a=2ba = 2b -> 2b+b=572b + b = 57