what is direct variation
Direct variation is a relationship between two variables where one is always a constant multiple of the other, so they increase or decrease together in the same proportion.
What “direct variation” means
When two quantities xxx and yyy are in direct variation:
- Their relationship can be written as
y=kxy=kxy=kx
where kkk is a constant called the constant of variation.
- If xxx doubles, yyy also doubles; if xxx is cut in half, yyy is cut in half, and so on.
- The ratio yx\dfrac{y}{x}xy is always the same number kkk for every pair of values.
So “what is direct variation?” → It’s a special proportional relationship where one variable is always a fixed multiple of the other.
Key facts (Quick Scoop)
- Equation form: y=kxy=kxy=kx.
- Constant of variation: k=yxk=\dfrac{y}{x}k=xy (and it does not change for that relationship).
- Graph: A straight line through the origin (0,0)(0,0)(0,0).
- Zero point: If x=0x=0x=0, then y=0y=0y=0; there is no separate +b+b+b term like in y=mx+by=mx+by=mx+b.
Example:
If you earn 15 dollars per hour, then
pay=15×hours\text{pay}=15\times \text{hours}pay=15×hours
This is direct variation with k=15k=15k=15: every time hours double, pay doubles too.
Common examples
- Buying fruit: total cost varies directly with weight (same price per unit each time).
- Distance at constant speed: distance varies directly with time if speed stays the same.
- Wages at fixed hourly rate: earnings vary directly with hours worked.
How to tell if it’s direct variation
- Check if you can write the rule as y=kxy=kxy=kx with a single constant kkk.
- For data pairs (x1,y1),(x2,y2),…(x_1,y_1),(x_2,y_2),\dots (x1,y1),(x2,y2),…, compute yx\dfrac{y}{x}xy for each; if all ratios match, it’s direct variation.
- On a graph, the line must be straight and pass through (0,0)(0,0)(0,0).
Mini comparison table
Here’s a quick comparison to keep the idea clear:
| Feature | Direct variation | Not direct variation |
|---|---|---|
| Basic equation form | $$y = kx$$, $$k \neq 0$$ | [5][3][9]$$y = mx + b$$ with $$b \neq 0$$, or non‑linear equations | [10][3]
| Ratio $$y/x$$ | Always the same constant $$k$$ | [7][1][3][9]Changes from pair to pair |
| Graph | Straight line through origin | [3][7][10]Line not through origin, or curved |
| Example | Cost = price × quantity | [5][7][9]Phone bill with base fee + per‑minute charge |
Tiny story to remember it
Imagine a snack stall that always charges the same price per cookie.
1 cookie, 3 dollars; 2 cookies, 6 dollars; 5 cookies, 15 dollars.
No matter what you buy, “money ÷ cookies” is always 3 — that fixed “3” is your
constant of variation kkk. The more cookies you get, the more you pay in
perfect proportion, and that’s exactly what direct variation is about.
TL;DR: Direct variation means yyy is always equal to some constant times xxx, written y=kxy=kxy=kx, so they grow and shrink together in the same ratio.
Information gathered from public forums or data available on the internet and portrayed here.
