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Chapter 6

Your Holiday Shopping List Strategies for Solving for X

H

olidays are a great time. We get to shop for our family and friends, and it’s always fun to pick out things we know they’ll love. But it’s so easy to spend more than we planned; those numbers sure do add up fast. Never mind the fact that the cutest shoes somehow ended up in our shopping cart. Now, how’d those get there? But seriously, it can be hard to figure out how to stick to a budget, even if we don’t splurge on ourselves. Later in this chapter, I’ll show you how to make your very own custom algebra formula that will totally keep you on track for the holidays . . . as long as you stay away from those shoes. We’ll also look at some new strategies for solving for x that you probably didn’t do in pre-algebra, and we’ll use the skills we’ve been building from the last few chapters to do it. First, here’s a mini review of the whole concept of solving for x to dust off the cobwebs in case it’s been awhile since you’ve thought about this stuff.

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Mini Review* of Solving for x

When a math problem asks us to find the value of x, our goal is to isolate x by doing things to both entire sides of the equation—so that we keep the scales always balanced— until x is all by itself on one side, and a number is on the other side. That number is the answer!

When isolating x, it’s helpful to think about inverse operations† . . . and gift wrapping.

Wrapping:

Unwrapping:

1. Put in box 2. Wrap with paper 3. Stick on sparkly bow

1. Unstick sparkly bow 2. Unwrap paper 3. Take out of box

Notice how the last thing we did to wrap the present is the first thing we undo to unwrap the present. Let’s wrap up x! If we first multiply x times 2, we get 2x. Then if we add 3, we get (2x + 3) , 2x + 3. Then if we divide the whole thing by 5, we get 5 right? So to isolate x, we unwrap it by first doing the inverse of the last thing we did (dividing by 5), which is multiplying by 5, and we get 2x + 3. Next, we do the inverse of adding 3, which is subtracting 3, and we get 2x. Lastly, we do the inverse of the first thing we did (multiplying by 2), which is dividing by 2, and we get x, totally unwrapped!

* For a more in-depth review of the basics of solving for x and “keeping the scales balanced,” check out Chapter 20 in Math Doesn’t Suck. † Inverse operations are operations that undo each other. Examples are addition & subtraction, multiplication & division, and opening & closing a box!

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With equations, we use this same strategy to isolate x, but we must do those same operations to the other side of the equation, too, so that we can simultaneously unwrap x while keeping the scales balanced: (2x + 3) = 3 5 (5)

(2x + 3) = (5)3 5 2x + 3 = 15

2x + 3 – 3 = 15 – 3 2x = 12 2x 12 = 2 2

x=6 And we end up with the answer, which we can plug back into the left-hand side of the original equation to check our work. It’s good to keep in mind that when we unwrap x, we’re undoing PEMDAS.* Notice that in this case, we first undid the only operation that wasn’t in the parentheses, followed by addition/subtraction and finally multiplication/division. Yup, PEMDAS in reverse.

Note: This was a lightning-fast review of pre-algebra solving for x, inverse operations, and undoing PEMDAS. From this point on, I’m going to assume you’re comfortable with solving these kinds of equations from pre-algebra. But if this chapter so far hasn’t been totally review for you, then I highly recommend reading Kiss My Math, especially Chapter 12, which has a ton of great pre-algebra strategies that get used constantly in algebra.

Solving for One Variable in Terms of Another Sometimes we need to solve for one variable “in terms of” another variable; for example, someone could ask us to solve for x in terms of y :

* See p. 400 in the Appendix for a review of the PEMDAS . . . and pandas! 78

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2y +

x = 5 3

And this can be weird, because when we isolate x, it’s usually just numbers that we’re isolating it from, right? So let’s make it nice and use flowers , just to take the edge off. and smiley faces 2A +

B 3

= 5

Let’s say we’re told to solve for the smiley face in terms of the flower. That means we need to isolate the smiley face: We’ll just blindly pretend the flower is any other number and collect it with the other numbers like we from both sides, we get: normally would. By subtracting 2

B 3

And to isolate

= 5 – 2A

, we multiply both sides by 3: 3

( B3 ) = 3(5 – 2A )

S

B = 15 – 6 A

And we’re done! So what does this mean? It means that if I tell you what number the flower is, you can instantly tell me what the smiley is. If I say, “The flower is equal to 0.5,” then you could quickly plug in 0.5 where and get = 15 – 6(0.5) = 15 – 3 = 12. And that’s what you see equals when is 0.5. We can also solve for the flower in terms of the smiley face. Now, we could start over with the original equation if we wanted, but the one we ended up with was so much nicer looking, and it’s also a true statement. In fact, it’s the same exact equation, just written differently! So, isolating = 15 – 6 , we’ll subtract 15 from both the flower in the equation sides and then divide both sides by –6:

B – 15 =

–6A

S

B – 15 –6

=

A

–1 Notice that if we multiply the fraction by , we can make it look –1 nicer: –1(B – 15) 15 – – B + 15 = = 6 6 –1(– 6)

B

15 – B , and we’re done! 6 Turns out, we can solve for anything in terms of . . . anything.

That means:

A=

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Heck, let’s go back to the equation, B = 5 – 2 A , and solve for 5 in 3 terms of the flowers and smiley faces. I mean, why not? It’s a free country. And we know what to do: To isolate 5, we’ll need to add 2 to both sides, right? And we get:

B 3

+ 2 A = 5, in other words: 5=

B+2 3

A

I admit, it’s kinda weird, but I wanted to show you that there are tons of ways to write the same true statement, and no matter what you’re asked to solve for, you can do it. It’s all a matter of perspective: Who’s the special variable? And everything else is just “stuff” that we need to move out of the way so we can isolate our very special variable.

Doing the Math a. Solve for y (in terms of x) and then evaluate at x = 0, 2. b. Solve for x (in terms of y) and then evaluate at y = 1, 9. I’ll do the first one for you. 1.

x + 2y = 6 + 3x 3

Working out the solution: For part a, we shall remain calm and just focus on getting y by itself. So, let’s multiply both sides by 3, and we’ll get x + 2y = 18 + 9x, right? Now, subtracting x from both sides, we get 2y = 18 + 8x. Finally, we’ll just divide both sides by 2: y = 9 + 4x . To finish part a, we’ll plug x = 0 into this last equation, and we get y = 9. When x = 2, then y = 9 + 4(22) = 17 17. We’re done with part a! For part b, we don’t have to start from the beginning; we can use our underlined equation above: y = 9 + 4x , which already looks so much simpler. To isolate x, let’s subtract 9 from both sides and then divide by 4 to get:

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y–9 y = 9 + 4x ◊ y – 9 = 4x ◊ 4 = x, which is y–9 the same thing as x = 4 . To finish part b, let’s plug in the values of y that we were given: If y = 1, then 9—9 0 x = 1 –4 9 = –8 4 = – 2. If y = 9, then x = 4 = 4 = 0 . y –9 Answer: a. y = 9 + 4 x ; 9, 17 b. x = 4 ; -2, 0 2. y + 4x = 2 3.

x+1 y = 2 (Hint: Start by multiplying both sides by y.)

4.

x + y = x (Hint: Start by multiplying both sides by 13.) 13

(Answers on p. 403)

It’s All About Perspective W

hen you and your sister or brother (or friend) get into arguments, have you noticed how you can deal with it better once you’ve taken the time to see the situation from the other person’s point of view? Solving for one variable in terms of another is like looking at an equation from one point of view, and then another. Solving for one variable in terms of another will come up constantly in word problems and graphing lines later in this book. It’s a great skill to have—for math and for siblings!

Before we move on to some strategies for more challenging problems, I’d like to share a little holiday joy. CHAPTER 6: YOUR HOLIDAY SHOPPING LIST

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Reality Math Your Holiday Shopping List: A Savvy Shopper Trick

M

aybe your parents still buy presents for people “on your behalf,” but soon enough, you’ll be making your own shopping list, saving up your allowance, and buying presents for friends and family yourself. It’s actually a really satisfying feeling! Your list might be shorter or longer than this, but you get the basic idea. Let’s say that after you’ve donated some toys to underprivileged kids, you’ve got $150 total left to spend on family and friends. Sounds easy enough, but problems start when you see the cutest dress for your BFF that costs $45, and then almost a third of your budget is gone. The next thing you know, you’re making your sister a life-sized bird out of toothpicks. Ah, just what she wanted. So . . . we’re going to make a custom, designer formula to help you stay on budget and in control of how you distribute your money! The first thing you need to do is to decide the person you’re going to spend the least amount of money on—like, say, your cousin. Sounds a little cruel, but c’mon, no one has to know . . . We’ll call that amount “x.” How much more than that do you want to spend on each other person on the list? You might think to yourself, Hmm, I’d probably want to spend about $10 more on my BFF than on my cousin. Then the amount you’d spend on your BFF would be x + 10. And maybe Mom will be x + 20. See where I’m going with this? Just fill in each person’s amount, in terms of x. Since we know the total equals $150, we can make a custom equation, and solve for x! x + 20 + x + 20 + x + 15 + x + 10 + x = 150

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Now, let’s combine like terms and solve! S 5x + 65 = 150 S 5x = 85 S x = $17 And now, to find out how much you can afford to spend on each person, just plug in 17 wherever you see x on the list. And voilà! A list that adds up to exactly your budget, $150, with easy guidelines for how much to spend on each person. Of course, if you find something perfect for your cousin that costs $20, you can always subtract $3 from your BFF, or maybe you get an amazingly cute top for your mom that only costs $34. You’ll end up adjusting the numbers in little ways as you shop, but it’s so helpful to have this to gauge how you’re doing. Then you’ll know exactly how to adjust along the way. Learning how to stay on budget for anything is a great skill, and algebra puts you in the driver’s seat.

Strategies for Solving for x Sometimes teachers throw some pretty complicated-looking equations at us to solve. So I’ve compiled a list of helpful strategies for those little suckers. Let’s tackle those problems that always seemed too scary or daunting before. You won’t be easily intimidated, not as long as I have anything to say about it, girl. Bring ’em on!

More Solving for x in Terms of Another Variable Let’s say we’re asked to solve for x in this equation. 3x x + ax = 1 – 4 2 Hmm . . . fractions as coefficients are always sort of scary, and that a is a bit annoying, isn’t it? Well, let’s get rid of the fractions by multiplying 3x –

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both sides by 4.* Let’s use the distributive property (say hi to everyone at the party!) and watch those negative signs:

(

4 3x –

x + ax 2

)

(

= 4 1 –

()

3x 4

)

( )

S 4(3x) – 4 x + 4(ax) = 4(1) – 4 3x 4 2 S 12x – 2x + 4ax = 4 – 3x S 10x + 4ax = 4 – 3x

Okay . . . this looks better. The next step is to gather all the stuff with x to one side, right? I know, I know—we still have “Mr. a” hanging around; that’s okay—just trust me. We’ll add 3x to both sides, and we get: S 13x + 4ax = 4 Now, all the stuff with x is together on one side, and it’s as simplified as we can make it. Remember all that factoring we did in Chapter 3? We’re going to now factor out an x by using reverse distribution: Yep, x is pulling out of the party. 13x + 4ax = 4 S x(13 + 4a) = 4 We’ve separated the x from “Mr. a,” yay! And how do we finish isolating x? By dividing both sides by that stuff in the parentheses so that the x ends up by itself: x(13 + 4a) = 4 S

x(13 + 4a) 4 = (13 + 4a) (13 + 4a) S x=

4 13 + 4a

Done! And if we plugged that back into the original equation, it’d be messy, but we’d get a true statement. Just to recap: Our strategy here was to collect the x’s on one side. Then we used our new factoring skills (reverse distribution) from Chapter 3 to surgically remove the x from the a, so we could isolate and solve for x. We also practiced the strategies of multiplication and distribution to get rid of those fractions to begin with.

* Notice that 4 is the LCD of both fractions in the equation. That’s why it works! 84

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Speaking of fractions, here are some crazy ones like we saw in Chapter 5, and now we’re going to solve equations with them.

Dealing with Big, Crazy Equations At some point, you’ll come across something like this: 3x 5 x x = 5 + 3 x + 4 2 Totally crazy, right? Well, the key behind solving these kinds of wacky fraction problems is to break them into bite-sized pieces. We deal with the numerator and denominator separately. In fact, hmm, we’ve already dealt with this very fraction (see p. 67)! So using the same strategy, this equation simplifies into: x 4 = + 3 5 5 Ah, looking much better. I feel like getting rid of the fraction element completely, don’t you? Let’s multiply both sides by 5: 5

( 45 ) = 5 ( 5x + 3)

S 4 = x + 15 And now we just subtract 15 from both sides and get x = –11. Answer: x = –11

“I like to think of x as an unknown ‘kiss’ from some guy and I’m solving to find out who (or what number) he is, so I have to get rid of all other numbers (distractions) to figure out his true identity!” Erin, 14

Summary of “Solving for x ” Strategies When solving for x (or any problem in life!), it’s not always so clear what to do next, is it? Well, you’re growing up, and so is the math: There will CHAPTER 6: YOUR HOLIDAY SHOPPING LIST

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often be many ways to tackle these problems. Below is a summary of my favorite ways to handle some of the sticky issues that come up in equations, many of which we’ve seen in the last few pages. But every problem is different, so use these tools when it makes sense to you!* And remember, above all, our goal is to isolate x while keeping the scales balanced.

Complaint

Possible Strategy

The coefficients are fractions. I don’t like it when the coefficients are fractions.

Use the distributive property to multiply both sides by the LCM of the denominators, so the fractions go away.

I don’t like all these parentheses!

Use the distributive property to multiply stuff out.

What’s with all the negative signs?

Multiply both sides by –1, and use the distributive property to do it right!

Um, one (or more) of the x’s is stuck to another variable.

Gather all the terms with x in them, and factor out the x. (See pp. 83–84.)

Hey, there’s an x on the bottom of a fraction. Help?

Multiply both sides by the denominator. Warning: You must check your answer, and denominators can’t equal 0!

The coefficients have decimals in them, and it’s kind of freaking me out.

Multiply both sides by 10, or 100, or 1000, etc. (We’ll practice this in Chapter 16.)

There is a certifiably crazy fraction on one side of this equation.

Before even starting to solve, simplify the numerator and denominator separately. Then create a simple fraction from it, and then move on to the solving part!

* Also check out the “Solving for x Workshop” on pp. 183–87 in Kiss My Math.

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Watch Out! Every time that I say “multiply both sides by” something, remember to use parentheses and the distributive property. It’s important that we multiply both entire sides by the same thing to keep the scales balanced. Otherwise, we’d end up with an untrue statement, and when we were finally left with “x = something,” it would be untrue, too!

Let’s lift some heavy weights and use the above strategies to solve these crazy-looking equations. Some will have other variables in them, and some won’t. Look, if you didn’t have to struggle with these at least a little, you wouldn’t be normal. It’s how we deal with fear that separates the strong young women from the little girls. Stick with me, sister, and you won’t be easily scared off!

Doing the Math Solve for x. For a challenge, try them without my hints! I’ll do the first one for you. 1.

ab + bx = 0.7x 1 a 2

Working out the solution: Even without knowing how this is going to turn out, let’s bring all the stuff with x together, so we’ll subtract bx from both sides, and get ab 1 a = 0.7x – bx. 2 On p. 54, we used complex fractions to learn that the left side equals 2b, so now our equation looks like this: 2b = 0.7x - bx. We can’t keep that decimal around, so multiplying both sides by 10 gives us 10(2b) = 10(0.7x - bx) ◊ 20b = 7x - 10bx. Looking better! To isolate x, let’s factor out x with reverse CHAPTER 6: YOUR HOLIDAY SHOPPING LIST

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distribution and write our equation like this: 20b = 7x - 10bx ◊ 20b = x (7 - 10b). To finish isolating x, we’ll divide both sides by (7 - 10b) and get 20b 7 - 10b = x . And that’s our answer! b Answer: x = 7 20 – 10 b 2. ax + bx = c (Hint: Factor out x first!) 3. x(y + z + 3) = 7(x + 2) (Hint: First distribute the x and 7, then collect the stuff with x’s to one side, and then factor out the x.) 4.

d(c – 2) + dx = 5 (Hint: First, simplify just the fraction. See (2 – c) pp. 48–49.)

5.

x x + 2 6 = 2x – 1 (Hint: First, simplify just the complex fraction 2 like we did on p. 67.) 3

(Answers on p. 403)

Takeaway Tips No matter how complicated things look, your goal is to isolate the variable you’re solving for. Collect those on one side and everything else on the other side. If x is stuck to other variables with multiplication, then after you gather the x’s to one side, factor out x with reverse distribution, like we learned in Chapter 3. Every problem is different! And practice is the best way to get good at solving for x.

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Danica’s Diary 50 THINGS THIS WEEK In the tenth grade, I used to worry about getting all my work done as well as everything else I had scheduled for the week. I was acting on the TV show The Wonder Years, so I had lines to memorize on top of all my school assignments and other commitments. But I wasn’t alone; I had friends on the school paper and sports teams who felt just as overwhelmed as I did——maybe more so! I was lucky to have a really great tenth-grade history teacher who seemed to know what we were all going through. Her name was Mrs. Hof, and she would often begin class by telling us a story about her life, usually having to do with issues of stress and time management. She was in the process of buying a house, selling her old house, teaching, grading papers, and driving her daughter to school and gymnastics, among other things. She told us that when she felt overwhelmed with too many things to do, she might think, “I need to get 50 things done this week, and I don’t know how to do 50 things in one week.” But then a moment later, she’d realize, “. . . but I can do these 10 things today. That much I know I can do. And you know what? I don’t have to worry at all about the other 40 things; I can really just focus on these 10 things for today. That’s it, 10 things.” I still think about that——all the time. When I got my first book deal, for Math Doesn’t Suck, I was really excited. I was going to be a real live book author! But then I was like, “Wait, a 300page book? Even if I have six months, I have no idea how to write a 300-page book!” And then I remembered Mrs. Hof’s method of breaking it down into smaller, bite-sized pieces of work, and I thought, “Hmm . . . let’s see. If it takes me about 3 weeks to write some sort of outline for it and brainstorm ideas, then I’ll have 21 weeks left to write 300 pages. That works out to just over 14 pages CHAPTER 6: YOUR HOLIDAY SHOPPING LIST

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a week. But let’s say I only work 5 days per week. Then I’d need to write 3 pages each day to be safe.” Just 3 pages? Well! That sure sounded better than 300 pages. Now some days I wrote 5 or 6 or even 10 pages if I was really on a roll. And some days I got busy with other things and didn’t get to write at all, or I spent the whole day rewriting pages so I didn’t add any new pages to the book. And whenever I felt panic creeping up, I’d think about Mrs. Hof and her strategy. And it worked! You can apply this strategy to just about anything, whether it’s a huge class history project or a long biology report, or even a multistep, scary looking math problem.* Don’t let the size of a task scare you off. That’s how a lot of people end up procrastinating and then cramming: They feel too scared to approach such a big endeavor; they’d rather hide from it, and then they end up with even less time, more stress, and a guilty conscience for procrastinating. The solution? Break it down into smaller pieces, and for now, focus only on the small piece in front of you. Don’t worry about the entire project, just the piece in front of you, and do that one piece now. It really works! As one of my ninth-grade teachers, Mr. Coombs, was fond of quoting, “The journey of a thousand miles begins with a single step.” I’m currently writing my third book, which is an enormous project. It’s easy to feel overwhelmed when there’s so much to do, but I’ve learned that breaking it down into bite-sized pieces and focusing on those one at a time makes all the difference. And if you’re reading this . . . then I guess it worked!

* To see what I mean about breaking down a math problem into bite-sized pieces, just see the problem on p. 68. 90

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Your Holiday Shopping List Chapter 6  

Your Holiday Shopping List Chapter 6

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