System of Equations Word Problem
by Bill Smith
(Atlanta, Georgia)
The telephone company offers two types of service. With Plan A, you can make an unlimited number of local calls per month for $18.50. With Plan B, you pay $6.50 monthly, plus 10 cents for each min. of calls after the first 40 min. At least how many min would you have to use the telephone each month to make Plan A the better option?
Karin from Algebra Class Says:
Hi Bill,
I really like this problem!
Since we are comparing two different plans, we must write an equation for each plan. When you write two different equations for the same problem, it is called a system of equations.
We will let y = the total bill and x = the number of minutes used.
Plan A: The fee is a flat rate. Nothing changes and the fee is not based on the number of minutes. Therefore, y = 18.50
Plan B: The cost is 10 cents per minute, after the first 40 minutes, plus 6.50.
y = .10(x-40) + 6.50.
We have 10 cents times the number of minutes minus 40. We have to subtract the first forty minutes because they are free). Then you must add on the monthly fee of 6.50.
Now let's simplify this equation a little further.
We need to use the distributive property.
y = .10x - 4 + 6.50
Now we can combine like terms (-4 + 6.50= 2.50)
y = .10x + 2.50
So our two equations are:
y = 18.50
y = .10x + 2.50
We know initially that Plan B is cheaper. If we solve the system and find the point where the two companies are the same price, then any minutes thereafter, Plan A will be cheaper.
So, let's solve the system. The best method to use is substitution.
Since y = 18.50, we can substitute this number for y into Plan B's equation.
y = .10x + 2.50
18.50 = .10x + 2.50
Step 1: Subtract 2.50 from both sides.
18.50 - 2.50 = .10x + 2.50 - 2.50
16 = .10x
Now, divide both sides by .10
16/.10 = .10x/.10
160 = x
Therefore, for 160 minutes, the two plans cost the same, $18.50. For any minutes over 160, Plan A would be the greater value.
Try it: Let's try 161 minutes.
Plan A = 18.50
Plan B = .10x +2.50
y = .10(161) + 2.50
y = 18.60
Therefore, Plan A is cheaper.
I hope this helps!
Karin