Exercise 5 Page 325

We are told that gift cards can be purchased for $10 or for $25. We want to know how many of each kind we could have purchased when spending $285.

Write the Equation

First, we need to write an equation in the standard form to model this situation. Let x be the number of $10 gift cards purchased and y be the number of $25 gift cards purchased. We get the following equation. 10x+25y=285

Possible Gift Card Combinations

We need to find three combinations for the numbers of each card purchased that fit our equation. These numbers need to be whole numbers because we cannot purchase a portion of a gift card. The best method here is to guess and check. But, we can make educated guesses! We have some things to consider when making these educated guesses.

• Think ahead about the steps we will take when solving the equation. For example, will we need to divide by the coefficient.
• In order to have a whole number for x, we need our y value to be an odd number. An odd number value for y ensures that, when solving for x, our right-hand side is divisible by 10. Let's solve for x when y=1.

See how the odd number for y gave us a number divisible by 10 on the right-hand side? Let's continue solving.

This means that if we bought 1 $25 gift card, we also bought 26 $10 gift cards. Now let's try substituting y=3 and y=5.

This means that if we bought 5 $25 gift cards, we also bought 16 $10 gift cards. While there are more possibilities, three example combinations are as follows.