I need help on 2 things based on this word problem.

Question

The student council is planning a party at the skating rink. There is a $100 fee to reserve the rink for the party. It will also cost $5 per person, which includes the rental skates and snacks.

a. I need to write an equation to represent the total cost of the party.

b. I need to identify the rate of change (slope), including units.

c. If this was graphed, what would be the y-intercept?

Answer 1

Recall the slope-intercept form of a linear equation is given by the following formula:

          y = mx + b ,

   where 'm' is the slope of the line and 'b' is the y-intercept.

a) Given that it will cost $5 per person in addition to a charge of $100 just to reserve the rink, we can conclude that the total cost of the party is the $5/person fee multiplied by the # of people attending the party plus the $100 reservation fee. That is,

     total cost of party = ($5/person)·(# of people attending) + ($100 reservation fee)

If we let 'y' represent the total cost of the party and 'x' represent the # of people attending the party, then

                            y = 5x + 100

b) Give the slope-intercept form of a linear equation:     y = mx + b

   And the equation we found to represent the total cost of the party:     y = 5x + 100

The slope (m) is equal to 5  ==>     $5/person     OR     5 dollars/person

c) Graphically, the y-intercept (b) is the point where the line intercepts with the y-axis (algebraically, the y-intercept is the point on the line when x = 0).

Thus, if you graph the line   y = 5x + 100 , the point at which the line crosses the y-axis is the y-intercept. When the line crosses the y-axis, the x-coordinate is thus 0. To find the y-intercept algebraically, plug in x=0 into the equation of the line:

     y = mx + b     ==>     y = m(0) + b     ==>     y = 0 + b     ==>     y = b

     y = 5x + 100   ==>   y = 5(0) + 100   ==>   y = 0 + 100   ==>   y = 100