The cost of your salad at the cafeteria VARIES DIRECTLY with the weight of your salad.
This relationship can be recognized from its "pattern".
C = k * w
C varies directly as w.
I'll give you an in-depth explanation, by detailing the salad-bar example. You need to read now. I'll be repeating key concepts.
First, let's define the three symbols above: C, k, and w.
The symbol C represents the salad's cost (the value of C is some amount of money that VARIES from one customer to another).
The symbol w represents the salad's weight (the value of w is some number of ounces that VARIES from one customer to another).
The symbols C and w are called "variables".
The symbol k is called "the constant of variation", and its value never changes because it IS a constant.
The constant in a direct variation must be a positive or negative Real number; it cannot be zero because C = 0*w means Cost = $0 (all salads are free), since there is no relationship with w at all in the equation C = 0.
In this example, the constant of the direct variation between C and w is the price-per-pound at this particular salad bar. Let's say that we live in the year 2016 and k = $7.45/oz. I'll use this price as k for calculating some related values of w and C in the blue list below.
So, this direct variation gives us a formula for calculating each salad's price. That's why we use symbols; the formula needs to be programmed into the computer that receives the weight (w) from the scale and generates the cost (C) on the display for all to see.
Cost = PricePerOunce * Weight
C = k * w
Direct variation always whittles down to this basic form.
One variable is the product of another variable times a constant.
The reason we call this relationship a "direct variation" is because C goes up when w goes up, and C goes down when w goes down.
In other words, when the salads get heavier, they cost more.
When w goes down (the salads weigh less), then C varies the same way (the salad's cost goes down).
We can see this by looking at a list of values for w and the corresponding values of C that result.
0.5 oz cost is $3.73
0.8 oz cost is $5.96
1.1 oz cost is $8.20
1.6 oz cost is $11.92
2.4 oz cost is $17.88
This is only one example of direct variation.
The direct-variation relationship always takes the same form:
y = k * x
Where y is a variable that represents something that varies the same way as x varies. x is also a variable. Changing x causes y to change directly -- if x goes up, so does y; if x goes down, so does y.
k is the constant of variation; it can be any Real number except zero.
One last note: the symbols C, y, k, w, and x are just names. The English alphabet provides 26 symbols in lower case and 26 symbols in upper case, to use for names. That's not enough symbols, so we use letters from other alphabets sometimes, too.
It matters not which names are used, in word problems. What MATTERS is that you pay attention to the difference between variables and constants, that you pay attention to any definitions given for their symbols (names), and learn to remember patterns that keep coming up over and over.
There are hundreds of billions of direct variations taking place on this planet every second. You'll be dealing with direct variation a lot.
Remember the pattern:
y = C * x
That's all the "teaching" I care to do, tonight, but I left room for some unsolicited opinion.
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