Math Corner: Solving for One Variable

From the another thread:

Do you guys know if anyone has written down rules for solving equations with variables? Here’s what I’m getting at: When you solve an equation with a variable, the sequence of actions you will take to isolate a variable. It seems like the first action you choose can be crucial, too. For example, look at this equation:

7(2e−1)−11=6+6e

What should you do first? I saw this at Khan Academy, and they used the distributive property first–getting “rid of” the 7 outside of the parenthesis. That does seem like the best move, but I didn’t realize that initially. I think I might have tried to add 11 to both sides first. And then my instinct might have been to divide by 7 on both sides, but I think that would have been a mistake. In working on several of these type of problems prior to this one, one of the important early moves involved “getting rid of” the number being multiplied by an operation with a parenthesis. For example,

200 = 20(4c+2)

I guess you could use the distributive property first, too.

In any event, I’m trying to find a list of rules or principles that will guide a student on the sequence of operations they should choose. What should be the first step, and then the second, third, etc. I did a quick search for something like this, but didn’t see anything.

12 thoughts on “Math Corner: Solving for One Variable

  1. Here’s a question I had about solving the following equation:

    1/9x-4 = z/3

    The first move I made was to add 4 to both sides, giving me

    1/9x = z/3+4

    Then I multiplied both sides by 9, giving me

    x = z/3+4*9

    Here’s my question: I have to do “z/3-4” first. That is, I can’t do “9*4” first. In other words I have to think like this “(z/3-4)*9” Right? If so, my question is, why is this? How do I explain this? Is it because when I added both sides by 4, that occurred before I multiplied both sides by 9? That would make sense, I guess. (In case it’s not clear, I’m trying to identify the rules or principles for solving problems like this.)

  2. Here’s another question. Let’s take the following equation:

    y = x+7*4

    What operation do I have to do first? Do I just follow the rules of operations?

    1st: operations in parenthesis
    2nd: exponents
    3rd: multiplication
    4th: division
    5th: addition
    6th: subtraction

    If you have multiple operations of the same type, then you start from left to right.

    I’m uncertain about this. A part of me feels like I wouldn’t see this form of an equation. Instead, the equation would be put into one of the two forms:

    4x+7

    or

    4(x+7)

    But these two equations are not equal and none are equal to

    x+(7*4)

    So which one is correct? And why?

    (By the way, is there any principle for the order of operations or is arbitrary?)

    1. I talked to Grace about to this. I believe she believes that the following is the correct equation:

      x+(7*4)

      However, let’s suppose that before we got to

      y = x+7*4

      the equation was

      y/4 = x+7

      and we multiplied both sides by 4, getting us to y = x+7*4. Then we couldn’t “7*4”. Grace explained this by saying that “y/4” and “x+7” are equal to each other, they have to be treated as one thing; you could enclose parenthesis around them to indicate this. Because of this, you can’t multiply the 4 to just the 7–you could multiply x by 4 and 7 by far, though (using the distributive property).

      But here’s a question: This doesn’t seem to apply when the numbers on one or both sides are all being multiplied or divided. For example, let’s take the following equation:

      y/4 = 4x*7

      We could multiply both sides by 4, and get

      y = 7x

      Why exactly can we divide the 4 being multiplied to the x by 4. That is, why don’t we treat “4x*7” in the same was as “4x+7?”

      (Note: I would guess the original question would look like “y/4 = 28x”, but I thought the form I choose would better show the difference between the original equation and this one.)

      1. But here’s a question: This doesn’t seem to apply when the numbers on one or both sides are all being multiplied or divided. For example, let’s take the following equation:

        y/4 = 4x*7

        We could multiply both sides by 4, and get

        y = 7x

        Why exactly can we divide the 4 being multiplied to the x by 4. That is, why don’t we treat “4x*7” in the same was as “4x+7?”

        Because multiplication in fractions is a cool thing. Remember how to multiply fractions: you can just multiply all the numerators and all the denominators.

        1/a * 2/b * 3/c * 4/d = 1*2*3*4 / abcd

        And because multiplication is commutative (you can multiply in any order and get the same answer), this means it’s also equal to 1*3*4*2/dbca.

        And this means 1/a * 2/b * 3/c * 4/d is the same thing as
        2/c * 4/a * 3/d * 1/b.

        So going back to your example

        (4x*7)/4

        is the same thing as
        4/4 * x * 7

        it’s also the same as 7/4 * x *4 but that’s icky (7/4 is an unpleasant number to work with, while 4/4 is just 1).

        4x*7 cannot be treated like 4(x+7) because they simply mean two completely different kinds of things. 4x*7 means 4 times x times 7: four of something, seven times. Four quarters for one football game, seven times means twenty-eight quarters.

        4(x+7) means something added to 7, plus something added to 7, plus something added to 7, plus something added to 7. Every team in the league expanded its roster by 7 players. How many players, all together, are on teams in the AFC West?

        Multiplication is commutative; addition is commutative. Multiplication and addition are not commutitative together.

    2. And this means 1/a * 2/b * 3/c * 4/d is the same thing as 2/c * 4/a * 3/d * 1/b.

      So let me test out my understanding. 1*2*3/a is the same as

      1/2*2*3,
      1*2/a*3, or
      1*2*3/a

      And I guess something similar would work in the following situation:

      1+2+3-a could be

      1-a+2+3,
      1+2-a+3, or
      1+2+3-a

      The same sort of thing is going on.

      (4x*7)/4

      is the same thing as
      4/4 * x * 7

      it’s also the same as 7/4 * x *4 but that’s icky (7/4 is an unpleasant number to work with, while 4/4 is just 1).

      And if my understanding is correct, the following is also another correction iteration:

      4*x/4*7

      4x*7 cannot be treated like 4(x+7) because they simply mean two completely different kinds of things.

      I understand they’re different. I’m still a little fuzzy as to why multiplication and adding, together are no longer commutative.

      1. Also, look at this:

        y=4(x+1)

        If I wanted to solve for x, I could divide both sides by 4, right? y/4-1 =x. (If I wanted to, I assume I could divide both sides by x+1 as well.

        If this is so this is a little confusing because multiplication and addition is involved. I’m assuming the parenthesis is what allows this.

        And y/4-1 is not equal to y-1/4, right?

  3. I’m a little under the gun on things, but the basic principal (I wouldn’t call them rules because equations take so many different forms that solving for a single variable calls for different strategies at different times — that’s why it’s math: it’s solving problems according to how they present themselves) is to express the equation in its simplest form and go from there.

    Basically, you’re looking for an expression where “like terms” are combined. If you have an x^2 over here and a 4x^2 over there, you want to combine them so it’s 5x^2.

    In your first example the 7(2e-1) makes that impossible, so distributing the 7 makes the most sense. But in your second example, because you can see that dividing by 20 turns the 200 into 10, a much more pleasant figure to work with, you should do that. If the 20 had been a 7 (or a 2.9 or a 7/8), you might take a different approach because 200 divided by 2.9 or 7/8 would be messy.

    The reason you don’t want a “rule” is that you’re already equipped with the rules. As long as you do the math according to the rules you already know, it doesn’t actually matter what you do first: it will all come out the same in the end. But if you divide by that 7 in the first example, you’re making things hard on yourself!

  4. but the basic principal (I wouldn’t call them rules because equations take so many different forms that solving for a single variable calls for different strategies at different times — that’s why it’s math: it’s solving problems according to how they present themselves) is to express the equation in its simplest form and go from there.

    First–is there a way we can define what “simplest form” means? Off the top of my head, the simplest form often (but maybe not always) involves the fewest numbers. It also involves put a number or equation into the easiest to understand or manipulate. But is there a way to define this? What makes a number or equation easy to understand and manipulate? Or are there too many novel situations to identify universal qualities?

    Second, by “rules” I’m thinking of something like a flow chart. If you see a certain form of an equation, generally, the first step should be (blank).

    In trying to teach this, my instinct is to identify this type of “if-then” rules–or at least help the student get more fluent in making these judgments. To me, it seems like the key to solving these type of problems. Doing the operations (e.g., adding, multiplying) is the easy part. It’s knowing what to do first. As you mentioned there are different ways of solving the problem, but if you can choose the most efficient and easy to understand and solve process, then you have an advantage.

    Basically, you’re looking for an expression where “like terms” are combined. If you have an x^2 over here and a 4x^2 over there, you want to combine them so it’s 5x^2.

    So would this be one of the first things a person should look to do?

    In your first example the 7(2e-1) makes that impossible, so distributing the 7 makes the most sense.

    I totally missed that, and I suspect I would have taken a long time to see the distribution avenue. This kind of experience is the impetus behind wanting to find “rules.” Or think of my endeavor as trying to find all the tools one could use to solve this sort of problem, and then building an “if-then” system. Using the distribution property as a tactic didn’t come to mind. I want to lay out all the different procedures, and then give my son various equations to use those procedures, in different sequences, in hopes his decision-making will improve. I just thought that if you could have an organized set of rules/principles, with sample equations (identifying certain patterns), this would be the most efficient and effective way of teaching and developing proficiency in this.

    But if you divide by that 7 in the first example, you’re making things hard on yourself!

    OK, but this is precisely the reason knowing which process to do first matters, don’t think you think?

    1. It is so hard to define simplification. But if I ask you how much I owe you for dinner and you say “seven times the amount Penny paid me, plus fifteen percent of your total for a tip,” I’m thinking you could make that a lot simpler for me.

      Sometimes you can’t give me all the info, so the simplest answer might be, “Twenty bucks plus whatever you want to throw in for Grace’s birthday.” That’s as simple as you might be able to answer my question given there’s an unknown.

      But fewest numbers is probably not the best way to think of it. Sometimes the simplest expression has more numbers. 7 + pi is two numbers, but 10.3141590 is one number, and 7 + pi is waaaay simpler.

      Basically you’re looking for (a) fractions in lowest terms (improper fractions are okay, but mixed fractions aren’t), (b) combined like terms, (c) parentheses only where they clarify, and…other stuff I can’t think of, like fractional exponents, radicals in the denominator…lots of stuff you pick up as you pick up more math.

      There might be a flowchart out there but I’ve never seen one. You might have to ask a more experienced math teacher than me. One guideline I might go with is that if the math starts getting icky, see if another strategy gives you nice numbers. But sometimes the icky numbers are what you have to work with. It happens.

      Do I think knowing what to do first matters? Not especially. As I said, if you do the math correctly it doesn’t matter, because the answer comes out the same. With practice, you learn to recognize at a glance that 20 divides nicely into 200, so you should begin there. But you totally don’t have to. If you distribute that 20 then combine like terms, you’re going to end up with the same answer.

      Also, at the developmental stage of algebra, I care more about the process (doing the math) than I care about the product (the answer). If you’re coaching young people in basketball and they do everything right but miss the shot, that’s a victory. And if they’re very young, just being able to run down the court, stand in the right spot, and not trip on the way is the victory.

  5. But fewest numbers is probably not the best way to think of it. Sometimes the simplest expression has more numbers. 7 + pi is two numbers, but 10.3141590 is one number, and 7 + pi is waaaay simpler.

    But are examples like this fairly uncommon? If not, then “fewest numbers” is not a good generalization for simplifying.

    Basically you’re looking for (a) fractions in lowest terms (improper fractions are okay, but mixed fractions aren’t), (b) combined like terms, (c) parentheses only where they clarify, and…other stuff I can’t think of, like fractional exponents, radicals in the denominator…lots of stuff you pick up as you pick up more math.

    I think this type of list is helpful, or moving to a place I’m looking for.

    (Question: why are mixed numbers not OK, but improper fractions are?)

    One guideline I might go with is that if the math starts getting icky, see if another strategy gives you nice numbers. But sometimes the icky numbers are what you have to work with. It happens.

    Here’s the conclusion I’m coming to: Words like “icky,” “simple,” “easy-to-understand”–to a large degree depend on experience. That is, after one solves a lot of these problems one gets a sense or feel for all of this. I’m trying to find a short-cut–namely, listing principles, coming up with “if-then” rules. Maybe there is no short cut, but if there is a way to make learning more efficient, that’s what I’m looking for.

    The alternative I face is having to generate or find practice equations and then give them to my son. This is hassle and not always easy to do. (There is an online website that gives math problems, but they don’t always provide the type of problems I’m looking for.)

    Do I think knowing what to do first matters? Not especially. As I said, if you do the math correctly it doesn’t matter, because the answer comes out the same.

    I feel like I’m someone who often struggles to find the most efficient, simplest way to solve a problem; I feel like I end up choosing one of the longer routes. Because of this, I feel like it really does matter. Think about this in the context of taking a test. Efficiency can matter at lot given a time limit, right?

    In the case of

    7(2e−1)−11=6+6e

    if I didn’t start with using the distributive principle, I think I would have hit several dead ends or got the answer wrong.

    Also, at the developmental stage of algebra, I care more about the process (doing the math) than I care about the product (the answer).

    Yeah, but don’t you think that’s what I’m doing. I’m focusing on the process–how to approach solving these problems, the principles you should use to solve them, and the way to determine the steps. That’s my main focus right now. I think if my son can get this, the other part will be a lot easier which isn’t to say he’ll get everything right. But he’ll be in way better shape if he can make good decisions when isolating a variable.

    1. Mixed fractions are not simplified because you can’t really do any mathematical operations with them. You have to convert them to improper fractions to do any adding or dividing or whatever.

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