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Systems of Linear Equations

Systems of Linear Equations

At a Glance - Solving Word Problems

Now that we've practiced turning words into linear equations, let's actually solve a couple of word problems. This is usually a three-step process:

  1. Find the linear equation being described.
  2. Figure out what question is being asked, and answer that question.
  3. Check your answer.

The fourth step, "take a nap," is totally optional.

Sample Problem

Jenna works at a retail shop. Yes, she still works there, even after all her thievery, but she'll tell you it has nothing to do with her old man owning the joint. She still makes $10 per hour, plus $3 for each item she sells.

  1. How much does Jenna make in one hour if she sells 5 items during that hour?
  2. How many items would Jenna need to sell in an hour to make $43 during that hour?

This word problem is describing a line with an equation we found earlier: y = 3x + 10, which describes the amount of cash (y) she makes in an hour if she sells x items.

Since we've found the linear equation, now we can answer the questions.

1. How much does Jenna make in one hour if she sells 5 items during that hour?

Since x is the number of items Jenna sells during one hour, we've got x = 5.

Then y = 3(5) + 10 = 25, which means Jenna would be paid $25. This amount doesn't include tips. Yeah, she makes tips, too. What can we say, this girl knows how to turn a buck.

2. How many items would Jenna need to sell in an hour to make $43 during that hour?

Since y is the amount Jenna is paid, if Jenna makes $43, we want to have y = 43. Plug that into our equation:

43 = 3x + 10

We can solve this equation for x to find:

Since x is the number of items Jenna sells during an hour, in order to make $43 Jenna must sell 11 items. Given her foolproof sales technique of breaking down into tears whenever someone decides not to buy something, she shouldn't have any problem hitting that mark.

Let's check that this is correct, though: If Jenna sells 11 items she'll make 3(11) + 10 dollars, which is indeed $43.

Sample Problem

Marcio spent $7 per day. Knowing Marcio, he probably spent it on Lotto scratchers. After five days, he had $8 left. How much money did Marcio start with?

First, we need to come up with a linear equation. The amount of money Marcio has depends on how many days have passed. Let's have x be the number of days that have passed, and y be the amount of money Marcio has.

The statement "after five days, he had $8 left'' tells us that the point (5, 8) is on the graph. It also tells us he "shockingly" hasn't struck it rich yet, or he probably would have given up on these silly things by now.

Since Marcio is spending $7 per day, the slope of the line is -7. We can use this information to find an equation for the line. Let's use point-slope form, since we have a point and a slope. We find the equation:

y – 8 = -7(x – 5)

Now we can worry about answering the question. The amount of money Marcio started with is the amount of money he had when 0 days had passed. Oh, to go back in time and have all that hard-earned cashola back, eh, Marcio?

We want to find the y-intercept of the line, since that's the point where x = 0 days. We can do this by rearranging our point-slope equation into slope-intercept form.

The y-intercept is 43, which means Marcio started with $43. Hey...that's how much Jenna made from selling her 11 items. These two might be in cahoots...

Let's make sure we're right. If Marcio started with $43 and spent $7 per day, after 5 days he would have 43 – 5(7) = 43 – 35, which is indeed 8 dollars.

Word problems that involve a linear equation can give us the information we need to write that equation in several different ways. We could be told two points on the line, or a point and a slope, or the y-intercept and the slope, or both intercepts. We could be given a treasure map that will lead us to the information we need, although those problems are more rare. Word problems can ask questions about the intercepts of the line or the slope. They can provide one coordinate of a point on the line and then ask for the other coordinate.

Come to think of it, they ask us for a whole lot of stuff without giving much back in return. We're in a one-sided relationship and should probably get out of it. We'll see what our therapist has to say about this on Tuesday.

After we find the line described by the word problem, the trick, as usual, is to figure out what the question is actually asking. Don't be distracted by any of its mumbo-jumbo.

Example 1

Antonio works for a car dealership. He's paid a base salary plus a commission for each car he sells. One year Antonio sold 10 cars and made $35,000. Another year he sold 13 cars and made $39,500. Gee, Antonio. Only 23 cars in two whole years? What are you doing, being nice and non-aggressive toward your customers? What kind of a car salesman are you?

What is Antonio's base salary, and what is his per-car commission?

Exercise 1

A dolphin eats seven fish per hour. How many hours until the dolphin has eaten 56 fish?

Exercise 2

Leslie sells pianos. She gets a base salary of $30,000 per year plus a commission for every piano she sells. Last year she sold 20 pianos and made $48,000. What is her per-piano commission?

Exercise 3

Giovanni starts 700 miles from L.A. and drives straight towards the city at 55 miles per hour. After how many hours of driving is he 370 miles away from L.A.?

Exercise 4

Anna made forty-one cookies and went to a party. She gave each kid 2 cookies, and had 3 cookies left over. She threw those out rather than eat them herself because she's trying to watch her girlish figure. How many kids were at the party?

Exercise 5

Robert lives due south of an amusement park. One day he left his home and drove due south, directly away from the amusement park. He couldn't take the screaming kids any more. After two hours, Robert was 95 miles from the amusement park. After five hours, he was 212 miles away.

Strangely, he could still hear them...

  1. How far is Robert's home from the amusement park?
  2. How fast was Robert driving?

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