Lesson 56: Identifying, Writing, and Graphing Direct Variation

New Concept

The equation $y = kx$, where $k$ is a nonzero constant called the constant of variation, shows a direct variation between $x$ and $y$.

What’s next

Next, you'll learn to identify this relationship in equations and data, write your own variation formulas, and visualize them on a graph.

Let's use direct variation to solve a real-world problem about earnings and hours worked. We will use the second key idea, Writing and Solving a Direct Variation Equation, to solve this problem.

Example Problem
The amount of money earned varies directly with the hours worked. If a person earns 90 dollars in 6 hours, how much will they earn in 10 hours?

Step-by-Step

1. First, we establish that the money earned, $y$, varies directly with the hours worked, $x$. The given point is $(6, 90)$.
2. Begin with the general equation for direct variation:

3. Substitute the values from the known point to find the constant of variation, $k$.

4. Solve for $k$ by dividing both sides by 6.

5. Now, write the specific direct variation equation for this relationship.

6. Use this equation to find the earnings for 10 hours by substituting $10$ for $x$.

7. Simplify to find the answer. The person will earn 150 dollars.

Property

An equation represents a direct variation if it is in the form $y = kx$, where $k$ is a nonzero constant called the constant of variation. The graph is a line passing through $(0, 0)$ and we can say '$y$ varies directly with $x$'.

Explanation

Think of direct variation as a perfect partnership! When one variable ($x$) changes, the other ($y$) changes by the exact same multiplier ($k$). If you work more hours ($x$), you earn more money ($y$) at a constant rate. It's a straight-line relationship that always starts at zero, making it fair and predictable.

Examples

• $y = 7x$ is a direct variation. The constant of variation, $k$, is $7$.
• $y = -3.5x$ is a direct variation. The constant of variation, $k$, is $-3.5$.
• The equation $y = x/5$ is a direct variation because it can be rewritten as $y = (\frac{1}{5})x$.

An equation might not look like $y=kx$ at first, but a little algebra can reveal its true form. We will use the first key idea, Identifying Direct Variation from an Equation, to solve this problem.

Example Problem
Tell whether the equation $y + 3x = 0$ represents a direct variation.

Step-by-Step

1. To determine if the equation represents a direct variation, we need to see if it can be written in the form $y = kx$.
2. Start with the given equation:

3. To isolate $y$, we subtract $3x$ from both sides of the equation.

4. The equation is now in the form $y = kx$. This confirms it is a direct variation, and the constant of variation, $k$, is $-3$.

Property

To check if a set of ordered pairs represents a direct variation, verify that the ratio $\frac{y}{x}$ is the same constant value for every pair (where $x \neq 0$).

Explanation

Are these points part of the same team? To find out, put each $(x, y)$ pair to the test by calculating the ratio $\frac{y}{x}$. If every single pair gives you the exact same number, then congratulations! You've found the constant of variation, and it's a true direct variation. One different ratio means it's a no-go.

Examples

• The set $(3, 12), (5, 20), (-2, -8)$ is a direct variation because $\frac{12}{3} = 4$, $\frac{20}{5} = 4$, and $\frac{-8}{-2} = 4$.
• The set $(2, 8), (4, 16), (5, 21)$ is not a direct variation because $\frac{8}{2} = 4$ but $\frac{21}{5} = 4.2$.