Identifying Direct Variation from Ordered Pairs
Identify direct variation from ordered pairs in Grade 9 algebra by checking that the ratio y/x is constant for all pairs, confirming the relationship y=kx passes through the origin.
Key Concepts
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 $\frac{21}{5} = 4.2$.
Common Questions
How do you identify direct variation from a table of ordered pairs?
Divide y by x for every ordered pair. If the ratio y/x is the same constant k for all pairs, the relationship is a direct variation y = kx. If even one ratio differs, it is not direct variation.
What must be true about the graph of a direct variation?
A direct variation graph is always a straight line that passes through the origin (0, 0). If the line does not cross at the origin, the relationship is linear but not a direct variation.
What is the constant of variation and how do you find it?
The constant of variation k is the ratio y/x in a direct variation equation y = kx. To find it, pick any ordered pair (not the origin) from the data and compute k = y ÷ x.