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Applications of One-Step Equations

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What if The Perfect Pizza charges $1.50 for a slice of pizza? However, the restaurant offers a$2.00 discount off the per-slice cost if you buy a whole pizza. A whole pizza costs $10. How could you find how many slices are in a whole pizza? After completing this Concept, you'll be able to solve real-world problems like this one. Watch This Guidance Let's use the skills we learned in the last two concepts to solve applications of one-step problems. Example A In the year 2017, Anne will be 45 years old. In what year was Anne born? The unknown here is the year Anne was born, so that’s our variable $x$ . Here’s our equation: $x + 45 &= 2017\\x + 45 - 45 &= 2017 - 45\\x &= 1972$ Anne was born in 1972. Example B A mail order electronics company stocks a new mini DVD player and is using a balance to determine the shipping weight. Using only one-pound weights, the shipping department found that the following arrangement balances: How much does each DVD player weigh? Solution Since the system balances, the total weight on each side must be equal. To write our equation, we’ll use $x$ for the weight of one DVD player, which is unknown. There are two DVD players, weighing a total of $2x$ pounds, on the left side of the balance, and on the right side are 5 1-pound weights, weighing a total of 5 pounds. So our equation is $2x = 5$ . Dividing both sides by 2 gives us $x = 2.5$ . Each DVD player weighs 2.5 pounds. Example C In 2004, Takeru Kobayashi of Nagano, Japan, ate 53.5 hot dogs in 12 minutes. This was 3 more hot dogs than his own previous world record, set in 2002. Calculate how many minutes it took him to eat one hot dog. Solution We know that the total time for 53.5 hot dogs is 12 minutes. We want to know the time for one hot dog, so that’s $x$ . Our equation is $53.5x = 12$ . Then we divide both sides by 53.5 to get $x = \frac{12}{53.5}$ , or $x = 0.224 \ minutes$ . We can also multiply by 60 to get the time in seconds; 0.224 minutes is about 13.5 seconds. So that’s how long it took Takeru to eat one hot dog. Watch this video for help with the Examples above. Vocabulary • An equation in which each term is either a constant or the product of a constant and a single variable is a linear equation . • We can add, subtract, multiply, or divide both sides of an equation by the same value and still have an equivalent equation. • To solve an equation, isolate the unknown variable on one side of the equation by applying one or more arithmetic operations to both sides. Guided Practice Calculate the following, using the problem from Example C: a) How many hot dogs he ate per minute. b) What his old record was. Solution: a) For this questions we’re looking for hot dogs per minute instead of minutes per hot dog, as in Example C. We’ll use the variable $y$ instead of $x$ this time so we don’t get the two confused. 12 minutes, times the number of hot dogs per minute, equals the total number of hot dogs, so $12y = 53.5$ . Dividing both sides by 12 gives us $y = \frac{53.5}{12}$ , or $y = 4.458$ hot dogs per minute. b) We know that his new record is 53.5, and we know that’s three more than his old record. If we call his old record $z$ , we can write the following equation: $z + 3 = 53.5$ . Subtracting 3 from both sides gives us $z = 50.5$ . So Takeru’s old record was 50.5 hot dogs in 12 minutes. Practice Peter is collecting tokens on breakfast cereal packets in order to get a model boat. In eight weeks he has collected 10 tokens. He needs 25 tokens for the boat. Write an equation and determine the following information. 1. How many more tokens he needs to collect, $n$ . 2. How many tokens he collects per week, $w$ . 3. How many more weeks remain until he can send off for his boat, $r$ . Juan has baked a cake and wants to sell it in his bakery. He is going to cut it into 12 slices and sell them individually. He wants to sell it for three times the cost of making it. The ingredients cost him$8.50, and he allowed \$1.25 to cover the cost of electricity to bake it. Write equations that describe the following statements.

1. The amount of money that he sells the cake for $(u)$ .
2. The amount of money he charges for each slice $(c)$ .
3. The total profit he makes on the cake $(w)$ .

Jane is baking cookies for a large party. She has a recipe that will make one batch of two dozen cookies, and she decides to make five batches. To make five batches, she finds that she will need 12.5 cups of flour and 15 eggs.

1. How many cookies will she make in all?
2. How many cups of flour go into one batch?
3. How many eggs go into one batch?
4. If Jane only has a dozen eggs on hand, how many more does she need to make five batches?
5. If she doesn’t go out to get more eggs, how many batches can she make? How many cookies will that be?