Question

Is the equation `6x^2+3y=0` an example of direct variation?

I think direct variation has something to do with `k/x=y`, but I'm not sure.

Answer 1

See explanation.

Explanation:

Two variables are in direct variation when the value of one is always a fixed multiple of the other. This means, for example, when one of the variables doubles, the other must also double.

If our variables are `x` and `y`, and `y` is always `k` times as much as `x`, then `x` and `y` are in direct variation. This relation is written as:

`y stackrel(color(grey)"is always")stackrel("")=k stackrel(color(grey)"times as much as")stackrel("") times x`

Here's an example: picture a fence. That fence's length can be measured in yards, but it can also be measured in feet.

Now, let's double the length of the fence. As a result, the number of yards in the fence has doubled, but so has the number of feet. This means the fence's length in feet is in direct variation with (or is directly proportional to) its length in yards.

This relation is quite easy to deduce. Because there are 3 feet in a yard, we have

`[("number"),("of feet")]= 3 times [("number"),("of yards")]`

or:`"        "y"          "= 3 xx"        "x`

What this equation says is: the fence's length in feet (`y`) is a constant multiple (3) of its length in yards (`x`).

`y` is always 3 times as much as `x`.

`y=3x`.

This equation could also be written as `y/x = 3`, to emphasize that the ratio between `y` and `x` is a constant 3. This is different than an equation of the form `k/x=y`; here, when `x` gets doubled, `y` gets halved. (Can you see why?) In this case, `x` and `y` are said to be inversely proportional.

Now, you were given the equation `6x^2+3y=0` and asked whether this was a direct variation. The answer is, quite simply, no. Let's solve this equation for `y`:

`6x^2+3y=0`

`color(white)(6x^2+)3y=0-6x^2"     "` (subtract `6x^2` from both sides)

`color(white)(6x^2+)3y="-"6x^2`

`color(white)(6x^2+3)y=("-"6x^2)/3"       "` (divide both sides by 3)

`color(white)(6x^2+3)y="-"2x^2`

There we go—we can see that `y` is directly proportional to `x^2`, but not to `x` itself.

Example: when `x=3`, we have `y="-"2(3)^2="-"2(9)="-"18`.
If we double `x` to 6, we get `y="-"2(6)^2="-"2(36)="-"72`, which is `4 xx "-"18`.
When `x` got doubled, `y` got quadrupled, so this is not direct variation.

Summary:

Long story short: you're right, if the equation can be rearranged to look like `y=kx` for some number `k`, then `y` and `x` are in direct variation. Otherwise, they're not.

Answer 2

if `x div y` gives the same answer, you have a direct variation.

`k = x/y`

Explanation:

A direct proportion (or variation) is a statement that two ratios are equal to each other.

`3:4 = 18:24`

If one quantity increases, the other increases in the same ratio.

`3/4 xx 6/6 = 18/24`

You will also know these as equivalent fractions - just a different name! for the same concept.

Consider the following.

`3/4 = 6/8 = 15/20 = 21/28 = 51/68 = 75/100 = x/y`

Any pair of these fractions represents a direct variation.

To decide whether two quantities are directly proportional, divide them.
Each of the fractions above gives the decimal `0.75`

So if `x div y` gives the same answer, you have a direct variation.

That "same answer" is what is called `k`.

So, the formula for a direct proportion is `k = x/y`

It is this concept which allows us to use cross-multiplying.

Examples of direct variations are:

• a straight line graph
• converting currencies
• a scale on a map
• buying more boxes and paying more money
• working out a distance from a given fuel consumption
• catering for guests