Assigning Variables
Transcript
To solve word problems algebraically, we need to introduce variables. The very first thing I will say is this,
be thoughtful in the choice of letter for your variable. It's true that algebra the math that we do does not depend on the letter at all.
尽管如此，您将使您的生活更轻松，并且您将通过选择有意义的字母来帮助自己。
选择一个变量将有助于，例如，从它所代表的人名称的相同字母开始。 Don't pick plain old x. This is a classic mistake that people make in word problems. 他们自动选择x,然后我将会发生什么s, they're a bunch of variables in the problem, they pick x for one of them.
They solve and then they get x and they're wondering, all right, well now I have this number, what is this number? Is this the answer or do I have to use this to find the answer? So they don't know when once they get the value, what it is that they've found. That's why it's much more helpful to pick letters that would represent the different items and different people in the situation.
Here's the beginning of a sample problem. I don't have the whole problem here, but here we have three women, Sarah, Mary, Helen. If I pick x for the salary of one of the women, I may correctly solve for it, but 当我得到x的号码时，我将如何知道它是否回答问题？我会得到那个x，然后我可能会想知道，现在，等等， 我只是找到了Sarah的薪水，或玛丽的薪水或海伦的薪水？
对于这个问题，S，M和H，这三个女性的每周工资将是更好的变量。 那样，如果我找到s，我确切地知道我找到了什么，那种东西。选择变量的其他考虑因素。 Sometimes it can be helpful to assign a variable for the smallest value and then express everything in terms of that.
Sometimes, it can be helpful to assign a variable for the target value, what the question is asking. And the advantage of doing this is if you choose the target value as your variable, then as soon as you find the value, you're done, you're done, 你回答了这个问题。此外，如果问题中的所有数量与单一数量有关， if everything is expressed in terms of a single quantity, make this conceptually central quantity the variable and then relate everything to that.
这是一个练习问题。暂停视频，然后我们会谈谈这个。 Okay, at Greg's Groceries, a half gallon container of organic milk costs $1.50 more than a half gallon container of orange ice cream. Eliot bought 7 half gallon containers of orange juice and 3 half gallon containers of organic milk, spending $27 for all 10 containers.
How much does a half gallon container of milk cost? And so, here we're looking for milk. So first of all, I'm gonna pick M for that. And that's my target value. 换句话说，如果我找到了m的价值，我已经回答了这个问题。现在注意到另一个变量，果汁的价格，好吧， 果汁比牛奶少1.50美元。
Milk is more, so juice is less, so J = M- 1.5. And so everywhere where I would have a J, I can just substitute M- 1.5, and I can write everything in terms of M. So, the total price would be, M, the price of milk, times 3, and then the price of juice times 7, and if we add up all those together, it should = 27.
So the price of juice times 7 + the price of milk times 3 = $27. So that's the single equation, and 我们现在所要做的就是解决了M的等式。我们要分发，并注意到那部分分配， 我们要乘以乘以7次1.5。让我们在一边思考这个，7次1.5。
嗯，1.5我可以将这个写成1 + 0.5。嗯，7次1是7。 And 7 times 0.5, 7 times a half, half of 7 is 3.5. Well now I just have to add those and I get 10.5. So it's 7 times 1.5 is 10.5, so we can use that. So now let's distribute, 我们得到7m-10.5 + 3m = 27。
Let's first combine the 7M + the 3M, combine the like terms there. We get 10M- 10.5 = 27, add 10.5 to both sides and 27 + 10.5 is 37.5. Now to solve for M, all we have to do is divide by 10. And of course, dividing by 10, we're just sliding the decimal place one place to the left, so M = 3.75.
或者换句话说，半加仑的有机牛奶容器+ 3.75美元，这是答案。 这是另一个练习问题，暂停视频，然后我们会谈论这个。好的，所以你可能会诱惑使用变量s，j和p， 但请注意，一切都在罚款方面。因此，即使这不是目标价值， 为罚款选择一个变量是有意义的，因为我们可以表达在罚款方面支付的内容。
So I'm gonna pick F for the fine. Pick F for the fine then we can express everything else in terms of that variable. 因此，我们被告知乔支付了3美元以上的罚款。所以，F超过4 + 3。 Peter paid $3 less than one-third of the fine, so F over 3- 3. And then Sam paid $4 less than half the fine, so that's F over 2- 4.
所以所有三个股票都应加起来整个罚款。换句话说，如果我们加入这个，那就是这样，他们应该加入F. So we'll just write that as a single equation. That's the equation for F. Just all the three contributions added, add up to the whole fine. Well, first of all, the +3 and the -3, those cancel.
所以我们这里有很多分数。如果我们摆脱分数，生活会更容易。 所以我们应该乘以一些允许我们清除分数的东西。而且，您可能会从分数上的课程中记住这一点， 如果我们乘以最不常见的分母倍数。分母的最不常见的倍数是12。
12 is the lowest number that's visible by 2 and by 3 and by 4. So we multiply every single term by 12. Well, 12 times F over 4 would give us 3F, 12 times F over 3 would give us 4F, 12 times F over 2 would give us 6F. And of course, 4 times 12 is 48 and then of course the F on the other side becomes 12F.
I just add these 3 + 4 + 6 = 13, add 48 to both sides, subtract 12F from both sides, 13f-12f只是f，所以f = 48.所以现在我们发现了f的价值。 We have found the value of the fine. But that's not the answer, that's not what the question is asking.
It's asking, how much did Sam pay? Well, Sam payed half the fine. 嗯，一半是24岁。他支付了4美元不到罚款。 Half is 24, so he paid $4 less, Sam paid $20. And incidentally, if you're curious, Joe paid $15 and Peter paid $13, and that's how they covered the fine.
In summary, do not use generic variables. Choose variables that will help you keep track of the numbers that in the problem. So the worst thing you can do is just pick x, because then you'll be wondering, once you find x, what did I find? 有时它有助于选择最小的值作为变量，有时它有助于选择目标值作为变量。
如果所有值与一个数量相关，则选择作为变量的中央数量。
Read full transcript
选择一个变量将有助于，例如，从它所代表的人名称的相同字母开始。 Don't pick plain old x. This is a classic mistake that people make in word problems. 他们自动选择x,然后我将会发生什么s, they're a bunch of variables in the problem, they pick x for one of them.
They solve and then they get x and they're wondering, all right, well now I have this number, what is this number? Is this the answer or do I have to use this to find the answer? So they don't know when once they get the value, what it is that they've found. That's why it's much more helpful to pick letters that would represent the different items and different people in the situation.
Here's the beginning of a sample problem. I don't have the whole problem here, but here we have three women, Sarah, Mary, Helen. If I pick x for the salary of one of the women, I may correctly solve for it, but 当我得到x的号码时，我将如何知道它是否回答问题？我会得到那个x，然后我可能会想知道，现在，等等， 我只是找到了Sarah的薪水，或玛丽的薪水或海伦的薪水？
对于这个问题，S，M和H，这三个女性的每周工资将是更好的变量。 那样，如果我找到s，我确切地知道我找到了什么，那种东西。选择变量的其他考虑因素。 Sometimes it can be helpful to assign a variable for the smallest value and then express everything in terms of that.
Sometimes, it can be helpful to assign a variable for the target value, what the question is asking. And the advantage of doing this is if you choose the target value as your variable, then as soon as you find the value, you're done, you're done, 你回答了这个问题。此外，如果问题中的所有数量与单一数量有关， if everything is expressed in terms of a single quantity, make this conceptually central quantity the variable and then relate everything to that.
这是一个练习问题。暂停视频，然后我们会谈谈这个。 Okay, at Greg's Groceries, a half gallon container of organic milk costs $1.50 more than a half gallon container of orange ice cream. Eliot bought 7 half gallon containers of orange juice and 3 half gallon containers of organic milk, spending $27 for all 10 containers.
How much does a half gallon container of milk cost? And so, here we're looking for milk. So first of all, I'm gonna pick M for that. And that's my target value. 换句话说，如果我找到了m的价值，我已经回答了这个问题。现在注意到另一个变量，果汁的价格，好吧， 果汁比牛奶少1.50美元。
Milk is more, so juice is less, so J = M- 1.5. And so everywhere where I would have a J, I can just substitute M- 1.5, and I can write everything in terms of M. So, the total price would be, M, the price of milk, times 3, and then the price of juice times 7, and if we add up all those together, it should = 27.
So the price of juice times 7 + the price of milk times 3 = $27. So that's the single equation, and 我们现在所要做的就是解决了M的等式。我们要分发，并注意到那部分分配， 我们要乘以乘以7次1.5。让我们在一边思考这个，7次1.5。
嗯，1.5我可以将这个写成1 + 0.5。嗯，7次1是7。 And 7 times 0.5, 7 times a half, half of 7 is 3.5. Well now I just have to add those and I get 10.5. So it's 7 times 1.5 is 10.5, so we can use that. So now let's distribute, 我们得到7m-10.5 + 3m = 27。
Let's first combine the 7M + the 3M, combine the like terms there. We get 10M- 10.5 = 27, add 10.5 to both sides and 27 + 10.5 is 37.5. Now to solve for M, all we have to do is divide by 10. And of course, dividing by 10, we're just sliding the decimal place one place to the left, so M = 3.75.
或者换句话说，半加仑的有机牛奶容器+ 3.75美元，这是答案。 这是另一个练习问题，暂停视频，然后我们会谈论这个。好的，所以你可能会诱惑使用变量s，j和p， 但请注意，一切都在罚款方面。因此，即使这不是目标价值， 为罚款选择一个变量是有意义的，因为我们可以表达在罚款方面支付的内容。
So I'm gonna pick F for the fine. Pick F for the fine then we can express everything else in terms of that variable. 因此，我们被告知乔支付了3美元以上的罚款。所以，F超过4 + 3。 Peter paid $3 less than one-third of the fine, so F over 3- 3. And then Sam paid $4 less than half the fine, so that's F over 2- 4.
所以所有三个股票都应加起来整个罚款。换句话说，如果我们加入这个，那就是这样，他们应该加入F. So we'll just write that as a single equation. That's the equation for F. Just all the three contributions added, add up to the whole fine. Well, first of all, the +3 and the -3, those cancel.
所以我们这里有很多分数。如果我们摆脱分数，生活会更容易。 所以我们应该乘以一些允许我们清除分数的东西。而且，您可能会从分数上的课程中记住这一点， 如果我们乘以最不常见的分母倍数。分母的最不常见的倍数是12。
12 is the lowest number that's visible by 2 and by 3 and by 4. So we multiply every single term by 12. Well, 12 times F over 4 would give us 3F, 12 times F over 3 would give us 4F, 12 times F over 2 would give us 6F. And of course, 4 times 12 is 48 and then of course the F on the other side becomes 12F.
I just add these 3 + 4 + 6 = 13, add 48 to both sides, subtract 12F from both sides, 13f-12f只是f，所以f = 48.所以现在我们发现了f的价值。 We have found the value of the fine. But that's not the answer, that's not what the question is asking.
It's asking, how much did Sam pay? Well, Sam payed half the fine. 嗯，一半是24岁。他支付了4美元不到罚款。 Half is 24, so he paid $4 less, Sam paid $20. And incidentally, if you're curious, Joe paid $15 and Peter paid $13, and that's how they covered the fine.
In summary, do not use generic variables. Choose variables that will help you keep track of the numbers that in the problem. So the worst thing you can do is just pick x, because then you'll be wondering, once you find x, what did I find? 有时它有助于选择最小的值作为变量，有时它有助于选择目标值作为变量。
如果所有值与一个数量相关，则选择作为变量的中央数量。
