solve a system of equations

Question

Mitchell is going to the water slides and needs to figure out which deal is better. He can pay $35 to go on the water slides as much as he wants, or he can pay $15 to get in, plus an additional $1 per trip down the water slides. If Mitchell goes on a certain number of trips down the water slides, the two options are equivalent in terms of cost. What is the cost?

Write a system of equations, graph them, and type the solution

Answer 1

The trick to dealing with mathematical word problems is to know how to parse the language as the necessary formula is often embedded in the text.

Let c be the total money Mitchell spends at the water park and let r be the total number of rides. In the first case, Mitchell pays a flat $35 for an unlimited number of rides. The flat cost is really the y-intercept of the equation. Since Mitchell can ride the water slide as many times as he wants after paying the $35 entrance fee, it is equivalent to saying the slope of the line is zero because Mitchell does not pay anything more for riding the water slide.

The second equation shall be left as an exercise following the logic above.

Once you have written equations for both options (use the variables c and r as given above) and set them equal to each other. Use the rules of algebraic manipulation to get anything having an r onto the same side of the equals sign.

When graphing, think about what your graph means. In this case, the total cost is the dependent variable and should be plotted on the vertical, or y-axis and the number of rides is the independent variable and should be plotted on the horizontal, or x-axis. Does it make sense for the number of rides to be negative? If not, then restrict the number of rides to being at least 0. Since you likely can't take fractional rides on a water slide, try all integer numbers starting at 0 and substitute into the equations you have written. You may stop graphing once you have two lines that intersect as that will be when the options have equivalent cost and you have solved the problem.