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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!


Comments for System of Equations Word Problem

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May 15, 2012
Systems of Equations Word Problem
by: Karin

I use Excel for basic data and formulas, but I'm not sure how or if it could be used in this situation. Unfortunately, I don't use Excel often, so I'm not familiar with all of it's functions. Most students taking Algebra 1 use a graphing calculator.


May 15, 2012
by: Anonymous

So how would you graph something like this on excel?

Oct 12, 2010
System of Equations Word Problem
by: Karin

HI Bill,

You can state your answer as an inequality.

x > 160 minutes.

The way the problem reads, I see it being set up as a system of equations. Since the question uses the word "atleast", you can use the inequality symbol (greater than) in your answer, as shown above.

I hope this helps,

Oct 12, 2010
different method of solving, please
by: Anonymous

I understand how you did this problem using system of equations, but my teacher wants it to be solved using inequalities, so can you please show me how to solve this equation using inequalities. Thanks!

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