Direct variation in math is a relationship where one quantity is always a constant multiple of another, usually written as y=kxy=kxy=kx, so when one changes, the other changes proportionally.

What is direct variation in math?

In math, direct variation (or direct proportionality) describes two variables xxx and yyy such that their ratio yx\frac{y}{x}xy​ is always the same constant kkk. This relationship is written as:

y=kxy=kxy=kx

where k≠0k\neq 0k=0 is called the constant of variation or constant of proportionality.

That means:

  • If xxx doubles, yyy also doubles.
  • If xxx is cut in half, yyy is also cut in half.
  • The graph is a straight line through the origin (0,0)(0,0)(0,0).

A quick real-life picture

Think of a fixed price per item, like apples costing 0.50 per apple.

  • 1 apple → 0.50
  • 2 apples → 1.00
  • 3 apples → 1.50

Total cost yyy varies directly with the number of apples xxx; the constant of variation is 0.50 per apple.

Mini breakdown: key facts

  • Definition : One variable is a constant multiple of the other.
  • Equation form : y=kxy=kxy=kx, with kkk a non-zero constant.
  • Proportionality symbol : You can first write y∝xy\propto xy∝x (“y is proportional to x”), then replace ∝\propto ∝ with kkk to get y=kxy=kxy=kx.
  • Graph : A straight line through the origin; no vertical intercept besides 0.
  • Behavior : Both increase or both decrease together by the same factor.

How to recognize direct variation

You’re usually looking for any of these clues:

  1. Equation looks like y=kxy=kxy=kx
    • No “+b+b+b” term, no exponents on xxx, just a constant times xxx.
  1. The ratio yx\frac{y}{x}xy​ is constant
    • Given pairs (x1,y1)(x_1,y_1)(x1​,y1​), (x2,y2)(x_2,y_2)(x2​,y2​), etc., check if
      y1x1=y2x2=y3x3\frac{y_1}{x_1}=\frac{y_2}{x_2}=\frac{y_3}{x_3}x1​y1​​=x2​y2​​=x3​y3​​, etc.
  1. Proportional word clues
    • Phrases like “varies directly as,” “is directly proportional to,” or “cost per item is constant” usually indicate direct variation.

Simple example

Imagine a car rental that charges 20 per hour:

y=20xy=20xy=20x

  • xxx = number of hours
  • yyy = total cost
  • k=20k=20k=20, the constant of variation (20 per hour).

If you double the hours, the cost doubles; if you triple the hours, the cost triples—that’s direct variation.

TL;DR: Direct variation is when two quantities are linked by y=kxy=kxy=kx so that they change together at a fixed ratio, giving a straight-line graph through the origin.

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