Once you have the equation put together, circle the unknown—the value you need to figure out. Then, write out your equation in a complete phrase. How much will it cost them in total?
Jo started playing a hot new game, but she only has two hours before she has to go out. She spent 20 minutes on level 1, 37 minutes on level 2, and 41 minutes on level 3. How much time does she have left to play level 4? This time you ild the whole equation. Com magnets solutions Math Magnets Solutions Below are some word problems and magnets. How much time does she have left for level 4? Equation Construction Now you build the equations!
Use the numbers that were given and x for the unknown. Cost per membership Figure these out from the problem statement. This means multiplication. The same thing for this problem - you fill in the x for the unknown.
Jo started playing a game that just came out, but she only has two hours before she has to go out. Com simplify, simplify Equation Construction solution Now you can build the equations! Jo started playing a game that just came out - but she only has two hours before she has to go out. How many does she have that can be played online? Does it make sense for her to buy the subscription? What we really care about here is what x is—the unknown number of games. In fact, we can get rid of that seven as long as we make sure we do the same thing to both sides of the equation.
If it helps, you can write it out, but just thinking about the equation in words first should be enough to help. An equals sign means that both sides are the same. So if we take 7 away from one side, we have to do the same thing to the other side of the equation:. We can take away subtracting 7 from7 games by of the equation. Com isolate your variables How does that help!
What you need is a way to use the operations that you already know addition, subtraction, multiplication, and division to solve equations.
The tricky part? You must preserve the equality. Equality means the same. When you do something to one side of the equation, you have to do the same thing to the other side of the equation. Knowing that your goal is to isolate the variable means that you know which numbers to move away from the left side.
So which operation do you use when? The opposite of addition is subtraction. So, if some number is being added on one side of the equation, and you want to move that number to the other side, you can subtract that number from both sides. The math term that describes opposite operations is inverse operations.
The basic math operations are addition, subtraction, multiplication, and division. An inverse operation is the operation that undoes an operation like addition undoes subtraction. This is why when you needed to get rid of the seven an added number , you subtracted seven from both sides. You need to use the inverse operation for the number to remove it. For a subtracted number, add. For a divided number, multiply, and so on. Whatever you decide to do to one side of the equation, you must do to the other.
That keeps the equation the same. There are other inverse operations out there. Can you think of other operation pairs that work? Head First: And welcome back to Algebra at Night.
So do you guys always travel in pairs? Head First: Ok, I think I get it—you can move numbers from one side of the equation to the other. Inverse Ops: Well, yes. Inverse Ops: Absolutely! A little addition here or multiplication there, and you can get almost any variable by itself. Head First: Ah, right. So addition is always paired with subtraction, multiplication always with division.. Head First: Very cool! So any last words before we sign off? Inverse Ops: Opposites attract, and multiplication is the opposite of division.
Head First: Same with addition and subtraction, right? Inverse Ops: Just a couple thoughts. You have to be careful that you keep the equation balanced. Until next time, may your multiplications always have a division, and your additions subtract. Inverse Ops: Well, not really go away—remember, our job is to keep everything in balance.
We just move things around. If you have a multiplication you need to move, you can undo that multiplication with a division—on both sides of the equation. Inverse operations help you isolate the variable. Below are equations that have unknowns and numbers on both sides of the equations.
Use inverse operations to isolate the variable and solve the equation. However, lots of other people did. They gave up on using x for multiplication and came up on with a few easier to read options: ean Each of these m n. Your job was to use inverse operations to isolate the variable and solve the equation. Check this one too Com r These are just two ot. Be careful, some of the names are used twice! Com Buy! She decided that she wants to get more games and not worry about the headset just yet.
I must have done something wrong - I thought I had enough money! Figure out where she went wrong. Figure out where Jo went wrong. Could she have prevented her mistake? The division was wrong! Cecking your work It also means using a specific technique called substitution. Substitution uses your solution in the original equation Substitution means putting something in for something else. A substitute teacher is in the place of a regular teacher, right? To check your work, you substitute in the answer you found for the variable in the original equation.
Substitution is a process that can be used not just for checking your work, but for other things too. Do it. Q: A: When else will we use substitution? Q: Why are there different notations for multiplication and division? Finally, multiplying a number by a variable is so common that just writing them next to each other is a lot less confusing than having a multiplication symbol in between. Q: When should I use parentheses versus dots versus just bumping the number and variable together?
If you have a number times a bunch of things, you can use parentheses. If you have a number times a variable, just push them together. As for the dot Q: too? Do addition and subtraction have other notations, A: Q: Nope, they stay the same. Plus means addition, and the minus means subtraction, but A: Q: As far as working with them, none. There seem to be a lot of elements that go into solving an equation, how do I keep track? Probably the easiest one to forget is checking your work Substitution means putting a new value back into the original equation.
In those cases, you can use the slash. Com need more video games Help her figure out the details! How much does she need to save up to buy all of these accessories? Make sure to check your work Check you work! How much will it cost for the new level? Now how much does Jo need to come up with?
She wants all the accessories and the new level for her game This one is pretty straightforward How many games does she need to sell to cover the new stuff? She wants all the accessories and the new level for the game How many does she need to sell to cover the new stuff? Use x or some other letter for your unknown.
Understand the problem statement Each problem will come with hints and an unknown. Think of it verbally first. Check your work!! Check that you have the right answer by plugging your number back into the original problem in place of the unknown. Make sure you always keep the equation equal by doing the same thing to both sides. If not, apply another technique to get the variable by itself.
After a trip to sell off 4 games and buy a headset, Jo signed into LIVE and bought that new level, and she is ready to play! This is the and with allsytstem she wanted, ready to go. Jo - ready gamer, anytimto take on any e!. Equationcross Take some time to sit back and give your right brain something to do.
The operation that undoes some other operation. Use this to do away with addition. Across Checking your work is really just Plug your answer back in to A variable, by itself.
Checking your work around is reallyto just Jiggling the equation solve Jiggling the equation around to solve it. Down 1. The anti-multiplication. Stands for an unknown. Down 4. Math sentences. Algebra is solving for Stands for anpromises. The key3. The key thing an equation promises. Checking your work is really just Com 2 more complicated equations Taking Algebra on the road I used to be so afraid of the unknown, but now Bob can take me anywhere Can you help Paul?
In fact, Paul wants to bring his buddies and really blow it out this weekend. But how many friends can he bring? Not only that, but there are a lot of costs to keep up with: Tickets ajama Paul - P 1 fan! Gas eath -lorida The big unknown in this problem is how many friends Paul can bring.
Paul can spend up to that amount on the trip. Does it matter how many people come? Com each guy costs something What else has to go into the equation to add up to the total cost of the trip? Gas is the same no matter how many guys are in the car. These three depend upon how many guys go. The costs are: gas, hotel, food, and tickets. Some depend on how many people come, but not all of them.
What we were expense of the mtissing is the rip itself. But what about food? Hotel rooms? Those all depend on how many guys come and Paul is one of those guys. And all of that has to be related to how much money Paul can actually spend.
The question is how do we figure out how much each guy costs? Com how much will it all cost? Costs Magnets Solutions Your job was to use the magnets below to put together what the cost of the trip is based on the number of guys coming. So this stays over here on the left. We need to multi a ticket, and a ple number of guys to figure out these costs by th ole thing is going to be.
You can replace the boxes below with actual numbers for each cost Use the specific numbers for the costs to figure out the actual equation for the trip. Back in Chapter 1, you worked equations to isolate the variable. We need to isolate the variable g in this equation to figure out how many guys can go on the trip. But where should you start? Or the The cost equation is a multistep equation. You have to deal with both the and the in order to solve for g.
Which should you do first? Will you get different answers if you do one before the other? Which is right? So you could first move the by using subtraction, if you wanted to, or you could deal with the g part first by using division. So the question is really, which is easiest to do first? We have to everything bdivide y Working with first was fine, but we were left with all those nasty decimal numbers to work with.
The same answer. Doing the subtraction first was easier, and we got the same answer. How are we supposed to take Paul, one other dude, and. What do you think we need to do here?
Com manipulate the equation Q: What about the order of operations? A: If you have a bunch of additions, subtractions, multiplications, and divisions all together that you need to work out, then yes, you need to follow the order of operations.
Q: Do I always have to solve my equations twice? How would I know which operation to do first on a problem? We just did that here to show that either approach works. As for what to do first with different problems, well, it depends. Then you have fewer things to deal with for later manipulations, like multiplication or division.
Q: Can there be even more complicated equations? What if there are lots and lots of steps? Q: When you start with a big long problem, do you have to write the problem out with words first? A: Not really. Writing out a problem also means you have to step back for a second and think about the context of the problem. Com more complicated equations I can bring 2. So, you technically got a correct solution for g, but 2. Whoa - hang on.
We can put 2 people in a room. But we can put two guys in each room. How do we fix this? What part of our equation has to change?
Is the cost for each guy still based on the number of guys? So nu e multiplied by th hotel fees. What are we supposed to do with those? Are they the same? If the same variable shows up more than once in an equation, the variable has to have the same value. A variable just represents a number in an equation. So each time g shows up, it must represent the same value. In fact, since g represents the same value each time it shows up, you can combine terms where g is the variable.
Com more complicated equations Below is the new trip cost equation. Use the new costs to figure out how many guys can go if they share a room. St tio ua simplifying the eq First, work inside the parentheses.
Then combine like terms. Next isolate the er Then solve for the numemb.. Really, how many guys can come? Can you figure out what it is? Com context is king Below is the new trip cost equation. Your job was to use the new costs to figure out how many guys can go if guys share a room.
So does this answer work? However, by putting two people in a room, now 4 people can come instead of just 2! Com more complicated equations Wow - much better. So it looks like 4 of us can go if we double up in the rooms.
Before I say anything to the guys, how can we be sure this is right? Always check your work! You need to check your work. Just plug your answer back into the equation wherever your variable appears, and make sure both sides of the equation come out to be the same. Check that you got the right answer by plugging your answer back into your equation and making sure things are equal. Com always check your work Check that you got the right answer by plugging it back into the equation and making sure things are equal.
Dude, can my girlfriend come? Q: Why did we plug 4. A: The numerical answer to the equation is 4. But, when we go back to check our work, we need to use the mathematically correct answer, 4. Q: Is it worth taking all that time to check your work?
A: Absolutely. It is crazy-frustrating to go through a whole problem and get an answer, but get it wrong. That is a totally preventable problem if you go back and check your work. Q: My work looks a little different than the solution, but I got the right answer. Did I do something wrong? Q: When we isolated the variable, how did you know to subtract before you divided? If you can reduce the number of terms in the equation by combining like terms, that usually makes things easier for later steps.
There will be fewer terms to deal with. By subtracting both sides, we could get rid of the altogether, and only have to divide one term by Q: A: What if I miss combining a term? No problem. As long as you manipulate the equation using the rules properly, you will not get a wrong answer Just go back to your original problem and start over.
Not at all. Q: How can this be math if there is more than one way to do the problem? For example, you could do away with multiplication and just use lots of additions. But using multiplication is another way to get to the same answer.
Com manage pajama death This is your chance to look at some problems, write some equations, and do some manipulating. They make most of their money on the road the music industry is a tough business. The deal Pajama Death has with the record company is that they get the same percentage of profit from all of the different revenue they make on tour. Touring Revenue 1. Com A couple of hints: a percentagine will be a decimal or fractionof an equation, and each type fit revenue will have the same pro percentage.
They are charged the same amount for each problem broken fan, bartender that resigned, etc. Figure out how much they were charged per incident. Concert Incidents 1. Com one problem, many approaches vv This is your chance to look at some problems, write some equations, and do some manipulating. Convert to a decimal, and then a percent. Com ing both Here we are div. You may not have divided by or added up all of the terms first, but they should all get you to the same answer.
Not only that, but girls might not cost as much as guys to bring I want to go too! We need another variable Since girls have a different cost associated with them, we need to treat them separately in our equation. Com there are girls now v Use the information about cost per guy and the information about cost per girl to write up the new equation we need to solve to figure out who can come.
They are We did these terms alrth e chapter. Tickets are the same for girls, but it same, guys or girls. Com Get all on one sidthe numbers e. Just think about how you worked with one variable. You put together a equation to say how things relate to each other in terms of the variable.
So with just guys, we were putting the cost of the trip in terms of the number of guys going, the g variable. Even the individual parts of the trip were that way, like how we figured out the cost of the hotel: How much the guys will be spending on hotels Great question.
For that, we need to talk a little bit more about what exactly a term is A term, on the other hand, is just a piece of an equation. But how do you figure out how many terms there are in an equation? Everything is glued together because everything is multiplied by 3. That makes substitution possible.
An equation in two variables establishes a proportional relationship between the two variables. A single equation in two variables can not be solved without an additional relationship. Term - A piece of an algebraic statement that is related by multiplication or division. Some equations have two correct answers, and some descriptions will be used twice! Com in terms of Com more complicated equations I think we should have the same number of girls as guys.
Once we say that, everywhere you see an r, you can substitute in a g since they are equal. Suddenly the equation looks a whole lot more solvable Finish up the road trip equation, and solve for g. Then solve for r. Yor job was to finish up the road trip equation, and solve for g. Q: Why did we only write part of the decimal, and not the whole thing? A: Because for this problem, we can only bring whole people. You could round down to the nearest whole person and just show two decimal places, but to avoid confusion, we used one decimal place.
Remember to always think about the context of the problem. The math might work out to 2. That makes sense, because justtota guys. It was part of the problem statement, and we just worked with it to solve our initial equation. What if you can only get an equation in terms of two variables? A: Depending upon your problem, that may be all you need.
The goal may just be the proportion of the variables and not a numerical answer. I have enough money to bring 4 of us. Road Trip! Pretty impressive. We have to agree on music to listen to. Com simplify these Simplify each of these mathematical expressions and combine like terms.
So remember that a term needs to be held together only by multiplication or division. Apply the operations ofrder of irst. Com more complicated equations Multicross Take some time to sit back and give your right brain something to do.
Plug Use Equations express a relationship 3. Equations express a relationship in Tells you what you're looking for and what you need to find it. A Tells youofwhat you're looking for and what together you needthrough to find it. A piece of an algebraic statement lumped together through 7.
Down Down 1. More than one variable. You Morecan than 4. In order to solve equation 3. You can do this toanlike terms.
In order to solve an equation, you have to do this to the 6. An absolute value equation typically has this many solutions. Com simplify the expressions Simplify each of these mathematical expressions and combine like terms.
Com more complicated equations Multicross Solution Take some time to sit back and give your right brain something to do. A piece of an algebraic statement lumped together through multiplication. You can do this to like terms. In order to solve an equation you have to do this to the variable. Sometimes you just gotta follow the stinking rules. But when it comes to Algebra, rules are a good thing. This new primetime hit pits two contestants against each other, struggling to solve math problems.
Good luck J This is This is Ka te. Mark up their work step by step to show how they each got their answer. Com who won this round? BE the judge solution Your job is to judge what Kate and Jack did.
Jack worked his entire problem from left to right. Who says? Kate solved the expression properly because she used the order of operations. She got the correct answer because she followed that order—which is really just a rule for working with numbers—precisely. Jack worked his problem from left to right.
Strangely menacing authority figure. Equations and expressions are written to communicate an order. Hello math chaos! That way, you can be sure you—and the Math or No Math contestants—are following the rules. The order of operations is one of the ways everyone can get the same answer to the same problem.
Then, do This is an exponent. Last, do You need to work through and get all of those pieces simplified first and write those down. Below are several problems that are partially worked out. Your job is to figure out what the next thing to do in the problem is.
What goes next? Com magnet solutions Math Magnets Solution Your job is to figure out what the next thing to do in the problem is. Since everything is grouped over the 2, you need to simplify inside the parentheses -and in there the exponent goes first. Addition e These magnets ar left over. Com rules for numeric operations Q: Where did the order of operations come from? Q: Do inverse operations always go together in the order of operations?
A: A: Q: Q: A: It was established by early mathematicians geeky people who do Algebra for fun who were trying to compare their work. The order let those folks talk to each other and get the same answers for their problems, which is a pretty big deal. Why did the order of operations get set up this way? A: The strongest operations go first. Finally, addition and subtraction. Q: A: Are roots exponents? Yes, which means that they go second in the order of operations.
If you need a refresher on the details, just turn to the appendix, where exponents and roots are discussed. They do. Addition, subtraction, multiplication and division are pretty straightforward. Exponents and roots are also inverse operations. Do I have to memorize this? Yes, but if you just think of the operations going in order of strength that should help. Q: Do I need to reduce fractions right away?
A: The fractions are up to you. Q: A: So are fractions really division, or can you leave fractions alone? If you want to divide your fractions to get a number like 0. Q: When can you drop the parentheses? Do they need to stay after you did what was inside? Q: A: This seems like a lot of steps.
Is it hard to keep track of all this? In fact, since Jack wrote down his steps, we were able to figure out where he made his mistakes, and why he got a wrong answer. You can keep track of what you did and check your work. Com back to the game The rules have changed a bit, too. So speed is definitely a factor. Rusty on fractions?
Just flip to the appendix in the back for a little help. Kate Jack Whose solution was fastest? Com 58 10 5 54 2 rules for numeric operations What? Kate totally blew off the order of operations, but got the right answer. There are properties as well as rules.
Kate used the associative and commutative properties to work with her equation, and then applied the order of operations. Properties like the associative and commutative properties are really just another type of rule Circle where you think she used a special property. Com re-group your equations You can re-group your equations The associative property lets you change the grouping of numbers in addition or multiplication operations. You can change the groupings of those numbers around all you want.
You can take a problem like 10 x 4. The Associative Property - Changing the groupings of a set of numbers being added or multiplied does not change the outcome of the operation. There are two expressions next to each other, regrouped for you. Do the answers come out to be the same? Are the answers the same? No Yes No Why did the answers come out the same for some of the problems and not for the others?
Com associative practice Your job was to solve both sets of problems, see if the answers came out the same, and figure out why you got the results you did. No vs. Yes No Regrouping vs. Com rules for numeric operations The associative property only works for addition or multiplication - NOT subtraction and division. This means you cannot regroup subtraction or division problems without changing the value of the solution.
You have to solve expressions with subtraction and division as written. Q: So do we need the order of operations or not? So what if I can move groupings around? A: The associative property means you can work through an expression in the easiest, fastest way. Grouping together fractions that are easy to work with saves tons of common denominator time, and you can do the same thing with decimals, too.
Grouping things in terms of how you want to work on them can sometimes help you get started on a tough problem, too! Q: A: Are there more properties? The commutative property lets you reorder items in an equation. Q: So the associative property lets me change the order of numbers, right? A: No - the associative property just says you can change the grouping of numbers that are added or multiplied.
However, all is not lost, There is a property that will help you out with ordering: the commutative property. Keep reading The associative property says you can change groupings in addition or multiplication, but NOT with subtraction or division!
Com re-order your problem It looks like Kate did more than just re-group things So is there a property that lets you move numbers around, too? The commutative property deals with the order of the terms in addition or multiplication operations. The commutative property says that you can add the numbers involved in addition operations or multiply the numbers in multiplication operations in any order and not affect the value of your answer. Com rules for numeric operations Who a m I?
Assume they always tell the truth about themselves. Fill in the blanks to the right to identify the attendees. What property was used? Com property roundup Who a m I?
Only the parentheses ciative. Theupsordstayed the same. Both the associative and commutative properties 0. Associative Property: Hi, commutative. Is everything all right? You look a little mixed up. Commutative Property: Ha, I get it. Mixing it up is my specialty. If you have some additions or multiplications, I can move the numbers around without causing any problems. Associative: Nice. I just work with parentheses. There are strict rules, though.
Commutative: Yeah, I have the same rules. Commutative: Same problem over here. Order is really important for subtraction and division, I guess, so I have to keep my hands off. Commutative: What? Associative: That both of us can be used outside of the order of operations without changing the answer. Commutative: Sure. We work any time! Commutative: It had its sentence commuted. Associative: No, what about it? Associative: Right.
Com the final round t round Jack got one point laslem right, for getting the prob being but Kate got 2 for right and fast. Not only that, but both Jack and Kate know about the order of operations and the commutative and associative properties. The Commutative Property er You can change the ord or ion dit ad h wit of the terms ging an ch ut ho wit n multiplicatio the results.
Com another win for kate? All that was left was to do the exponent and then add and subtract. Jack canc things outeled here. The exponent is next. Com Circle one Jack rules for numeric operations Are you kidding me?
Every time we figure out how to judge these things, Kate pulls out another trick. The distributive property lets you multiply over several numbers. Kate got rid of all the fractions in one step by multiplying all her fractions by That canceled her denominators. Kate multiplied each term inside the parentheses by Here, Kate simplified each fraction by he dividing out t denominators. The distributive property says that if you have two groups multiplied together, you can simplify the groups, then multiply; or multiply first and then simplify.
She multiplied through first, then simplified. The 12 outside the parentheses is multiplied with each number inside the parentheses. Just like with the associative and commutative properties, the distributive property is about working with problems in a simpler, more efficient way. And these properties work with the order of operations, not against them.
In that case, you can distribute the number over the grouping. Take the expressions below and simplify them in two ways: by distributing, and by following the order of operations. Which way do you think is faster for each problem? Which way was faster? Com distributive drill Your job was to take the expressions below and simplify them in two ways: by distributing, and by following the order of operations.
Which way did you think was faster for each problem? I think that the distributive way was faster since all the fractions went away. The order of operations was the faster way to go since the numbers worked out in the parentheses. Com rules for numeric operations The Commutative Property der You can change the or of the terms with addition or multiplication without changing the results. The As sociativ e Prope rty operty r P e v i t tribu s i D e h T ed by multipli ing same ber be A num produces the ing the fy p a grou either simpli ing, or y b ip ult ly result then m m within the d n a ter group.
How am I supposed to remember all this? First there was the order of operations, and now there are all these rules You can memorize some general equations, not a lot of text. Remembering all those sentences for each property is a pain Fortunately, we can turn these rules into some simple equations.
But to do that, we need a way to represent numbers But what exactly is a constant? The constant just represents a number. Constants are great for turning specific problems into more general ones because we can use letters instead of specific numbers. For example At first, it might seem like these letters are just as hard to remember as a bunch of sentences. This is a much easier thing to remember. This is any words! Com A general equation is just a way to remember rules that apply to all numbers in a certain situation.
Be careful, though Com which property? Some of the property names are used twice! Com rules for numeric operations The Commutative Property The Commutative Property n multiplicatio der You can change the or ion of the terms with addit or multiplication without changing the results.
Kate got done faster and smarter and could make the fractions much easier. Kate made fractions much easier by distributing. Jack was the runner up. Book Description. Having trouble understanding algebra? Do algebraic concepts, equations, and logic just make your head spin? We have Upload a Thing!
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