Direct Variation Equations

A direct variation is a special type of linear relationship where two variables are directly proportional. In this relationship, as one variable increases or decreases, the other does the same at a constant rate.

The equation for a direct variation is always written in the form: $y = kx$ where $k$ is a non-zero number called the constant of variation (or constant of proportionality).

Key Features of Direct Variation

Direct variation equations have a few unique characteristics that make them easy to spot:

• No y-intercept term: Unlike the general linear equation $y = mx + b$, a direct variation equation has a $y$-intercept of zero ($b = 0$).
• Passes through the origin: Because $b = 0$, the graph of a direct variation is always a straight line that passes exactly through the origin $(0, 0)$.

Identifying Direct Variation Equations

To determine if an equation represents a direct variation, check if it can be written strictly as $y = kx$ with nothing added or subtracted.

Example 1: Which represents direct variation: $y = 2x$ or $y = 2x + 1$?

• $y = 2x$ is a direct variation because it perfectly matches the $y = kx$ format, where $k = 2$.
• $y = 2x + 1$ is not a direct variation. The $+ 1$ means the line crosses the y-axis at $1$, not $0$.

Example 2: Is $y = 4x$ a direct variation?

• Yes, it is a direct variation. It is in the form $y = kx$ with a constant of variation $k = 4$.

Solving Direct Variation Problems

If a problem states that "$y$ varies directly with $x$," you can use a given pair of $x$ and $y$ values to find the constant $k$ and write the full equation.

Example 3: If $y$ varies directly with $x$, and $y = 12$ when $x = 3$, find $k$ and write the equation.

1. Start with the formula: $y = kx$
2. Substitute the given values: $12 = k(3)$
3. Solve for $k$: Divide both sides by $3$. $k = \frac{12}{3} = 4$
4. Write the final equation: Now that you know $k = 4$, plug it back into the general formula. $y = 4x$

Once you have the equation $y = 4x$, you can use it to find $y$ for any other value of $x$, or vice versa!