Putting It Together: Solving Linear Equations

At the beginning of the module, Joan had applied for a credit card with air miles so she could pay her regular bills with it. Her hope was to build up enough miles to take a trip. Joan has planned to use her air miles card for the following things each month:

  • gas = [latex]$30[/latex]
  • car insurance = [latex]$40[/latex]
  • mobile phone bill = [latex]$65[/latex]
  • groceries = [latex]$150[/latex]
  • entertainment – restaurants, movies, and live music = [latex]$50[/latex]

The total monthly expenses she plans to put on her card is [latex]$335[/latex].

Each dollar she spends gives her [latex]1.25[/latex] air miles, and she expects to need about [latex]1200[/latex] air miles to buy the ticket she wants. For signing up, the card gave Joan [latex]100[/latex] free miles.

Help Joan plan for her trip by figuring out how many months she needs to pay her bills with her card before she has enough air miles. Because she earns a fixed amount of air miles for each dollar spent, this is a good candidate for a linear equation.

Since we know the amount of air miles earned depends on the number of dollars spent, we can write an equation  for air miles with an input of dollars:

[latex]\text{Air Miles} =\text{Number of air miles earned depends on dollars spent}[/latex]

The initial amount of air miles Joan has is [latex]100[/latex], so taking this into account, our equation will look like:

[latex]\text{Air Miles}=100+1.25d[/latex]

Since Joan needs to earn 1200 air miles to buy the ticket she wants, set the equation [latex]\text{Air Miles}=1200[/latex].

[latex]1200=100+1.25d[/latex]

[latex]1100=1.25d[/latex]

[latex]880=d[/latex]

Joan needs to spend $880 before she has enough air miles to buy the ticket she wants. Divide $880 by Joan’s monthly expenses of $335 to determine the number of months Joan needs to pay her bills with her card before having enough air miles.

[latex]\dfrac{880}{335}\approx2.63[/latex]

Since the answer has to be in months and two months would not earn Joan enough air miles, Joan would need to pay her bills for three months with her card before having enough air miles to purchase her ticket.