SOLUTION: If y varies directly with x, and y = 280 when x = 400, find the constant of variation k.

Question 79810: If y varies directly with x, and y = 280 when x = 400, find the constant of variation k.

Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
Since y varies directly with x, you can write the equation:
.

.
In this equation the k is the constant of variation.
.
The problem then tells you that y = 280 when x is equal to 400. Substitute these two
values into your equation:
.

.
Now you can solve for k by dividing both sides by 400 to get:
.

.
The left side can be reduced by dividing both the numerator and denominator by their
common factor of 40 to get:
.

.
That's the answer for k. If you substitute that value for k into your equation, the
equation becomes:
.

.
Hope this helps you to understand the "constant of variation" (sometimes also called
the "constant of proportionality").
.
This problem involves "varying directly as ..." and this means that the equation is:
.

.
Notice that as x gets bigger, y gets bigger also. That's why they are said to vary
"directly".
.
Other problems may ask you to find the "constant of variation" for quantities that
vary inversely as ...
.
For problems involving "inversely as ..." the equation becomes:
.

.
You can then solve for k by substituting the given corresponding values of y and x, almost
the same as above. For example, if your original problem had told you that y varied
inversely as x and that when y = 280 then x = 400, you could find k by the following
steps:
.

.
Multiply both sides by 400 to get:
.

.
and the equation would become:
.

.
Notice that as x gets bigger, y gets smaller ... and that's why the two have a relationship
that varies "inversely".
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