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### Course: 7th grade (Illustrative Mathematics) > Unit 2

Lesson 3: Lesson 4: Proportional relationships and equations- Identifying the constant of proportionality from equation
- Constant of proportionality from equations
- Constant of proportionality from table (with equations)
- Constant of proportionality from tables (with equations)
- Equations for proportional relationships
- Writing proportional equations from tables
- Writing proportional equations from tables

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# Equations for proportional relationships

Learn how to write a proportional equation y=kx where k is the so-called "constant of proportionality".

## Want to join the conversation?

- So I am doing the practice problems for this right now and sometimes the constant of proportionality is a fraction, like "y=1/3x" but sometimes it is a number or a decimal, like "y=0.34x" or "y=4x". How do I know which one to do? There have been multiple times where I put the decimal equal to the fraction, like 0.33 for 1/3 and gotten it wrong because it was supposed to be the fraction (and vice versa)(16 votes)
- I have experienced similar issues entering answers. After doing quite a few of these types of problems, I have found that entering your answer as a fraction is the safer bet, especially when your answer is something like 10/7x=y as 10/7 is a repeating decimal. I enter the answer as a decimal only if the question prompts me to do something like "round my answer to the nearest hundredth." Then I obviously know my answer should have a decimal in it. Otherwise, I just seem to run into problems with entering them usually do to rounding.

So yeah, I just feel that it is better to answer these kinds of questions with a fraction instead of the decimal unless you are specifically told to do so in the question.(19 votes)

- exelente video me ayudo mucho(8 votes)
- Si tu lo dices/If you say so(4 votes)

- abooga woogy bippity bopity boo

math is ugly(7 votes)- mathematics is an interesting subject(2 votes)

- i dont understand this(8 votes)
- what he is trying to say when 4 and 1 I think they mean the unit rate is 4 or it can 4/1(0 votes)

- This video taught me what my math teacher has been trying to teach my class for half a month now in the span of 1 minute and 18 seconds. Thanks Khan Academy.(5 votes)
- where is the practice questions?(4 votes)
- what is real life example of the equation y=1/20*x(4 votes)
- why does y over x require a five minute video I think it requires like half of that time.(3 votes)
- So I am doing the practice problems for this right now and sometimes the constant of proportionality is a fraction, like "y=1/3x" but sometimes it is a number or a decimal, like "y=0.34x" or "y=4x". How do I know which one to do? There have been multiple times where I put the decimal equal to the fraction, like 0.33 for 1/3 and gotten it wrong because it was supposed to be the fraction (and vice versa)I have experienced similar issues entering answers. After doing quite a few of these types of problems, I have found that entering your answer as a fraction is the safer bet, especially when your answer is something like 10/7x=y as 10/7 is a repeating decimal. I enter the answer as a decimal only if the question prompts me to do something like "round my answer to the nearest hundredth." Then I obviously know my answer should have a decimal in it. Otherwise, I just seem to run into problems with entering them usually do to rounding.

So yeah, I just feel that it is better to answer these kinds of questions with a fraction instead of the decimal unless you are specifically told to(3 votes) - ?? does this mean in real life ?_?(3 votes)
- What about buying things at the grocery store. A can of beans costs .99. This gives a proportional relationship cost (c) and number of cans (n) gives a formula c=.99n. This allows you to tell you cost depending on how many you buy. If you are the store buying a large quantity of beans, there is probably lower costs per unit depending on the number of cans you buy (in the hundreds or thousands of cans). You can figure out your maximum profit based on buying the correct number of cans, you want to get close to the number you project to sell because if you buy too many and do not sell them all, you may have to discard them when they expire, so while you bought them cheaper, it may not be to your best advantage.(1 vote)

## Video transcript

- So, let's set up a relationship between the variables x and y. So, let's say, so this is x and this is y, and when x is one, y is four, and when x is two, y is eight, and when x is three, y is 12. Now, you might immediately recognize that this is a proportional relationship. And remember, in order for it to be a proportional relationship, the ratio between the two variables
is always constant. So, for example, if I look
at y over x here, we see that y over x, here it's four
over one, which is just four. Eight over two is just four. Eight halves is the same thing as four. 12 over three it's the same thing as four. Y over x is always equal to four. In fact, I can make another column here. I can make another column
here where I have y over x, here it's four over one,
which is equal to four. Here it's eight over two,
which is equal to four. Here it's 12 over three,
which is equal to four. And so, you can actually
use this information, the ratio, the ratio between
y and x is this constant four, to express the relationship
between y and x as an equation. In fact, in some ways this
is, or in a lot of ways, this is already an
equation, but I can make it a little bit clearer, if I
multiply both sides by x. If I multiply both sides by x, if I multiply both sides
by x, I am left with, well, x divided by x, you'd just
have y on the left hand side. Y is equal to 4x and
you see that's the case. X is one, four times that is four. X is two, four times that is eight. So, here you go, we're
multiplying by four. We are multiplying by four,
we are multiplying by four. And so, four, in this
case, four, in this case, in this situation, this is our
constant of proportionality. Constant, constant, sometimes people will say proportionality constant. Constant of proportionality, portionality. Now sometimes, it might even be described as a rate of change and
you're like well, Sal, how is this a, how would
four be a rate of change? And, to make that a little bit clearer, let me actually do another example, but this time, I'll actually
put some units there. So let's say that, let's say
that I have, let's say that x-- Let me do this, I already
used yellow, let me use blue. So let's x, let's say
that's a measure of time and y is a measure of distance. Or, let me put it this way, x
is time in terms of seconds. Let me write it this way. So, x, x is going to
be in seconds and then, y is going to be in meters. So, this is meters, the units, and this right over here is seconds. So, after one second, we have traveled, oh, I don't know, seven meters. After two seconds, we've
traveled 14 meters. After three seconds,
we've traveled 21 meters, and you can verify that this
is a proportional relationship. The ratio between y and x is always seven. Seven over one, 14 over
two, 21 over three. But, I wanna write that
in terms of it's units. So, y over x is going to
be, if we look at this point right over here, it's seven
meters over one second. Seven meters over one second, or it's equal to seven meters per second. If you look at it right over
here, if you say y over x, it's 14 meters, 14 meters, in
two seconds, in two seconds. Well, 14 over two is seven,
14 over two is seven, and then the units are meters per second. So, that's why this constant, this seven, in all of these cases we have
y over x is equal to seven, that this is also sometimes
considered a rate. And over here it's very clear,
this is my distance per time. Now, if I wanted to write it generally, I could say that, look, if I'm dealing with a proportional relationship, it's going to be of the
form, I can always construct and equation of the form, of the form, either y over x is equal to k, where k is some constant. In this first example, k was equal to four and in this second example,
k is equal to seven. Or, you can just manipulate
it algebraically, multiply both sides by x and
you would have y is equal to, y is equal to kx, where
once again k is our constant of proportionality or
proportionality constant. So, this is a really, in some
ways it's a very simple idea, but in a lot of ways, you'll
see this showing up multiple, many, many times in
your mathematical career and it's neat to be able to recognize this as a proportional relationship.