Direct Variation
Direct variation is a Grade 7 math concept from Yoshiwara Intermediate Algebra describing a relationship where y = kx, meaning y varies directly with x and the ratio y/x is always the constant k. Students learn to identify, write, and graph direct variation equations.
Key Concepts
Property $y$ varies directly with $x$ if $$y = kx$$ where $k$ is a positive constant called the constant of variation . If $y$ varies directly with $x$, we may also say that $y$ is directly proportional to $x$. This relationship defines a linear function whose graph is a straight line passing through the origin.
Examples The total cost, $C$, of concert tickets varies directly with the number of tickets, $n$, purchased. If each ticket is 50 dollars, the relationship is $C = 50n$. The distance, $d$, you travel at a constant speed varies directly with time, $t$. If you are driving at 60 miles per hour, the formula is $d = 60t$. The amount of interest, $I$, earned in one year is directly proportional to the principal, $P$, invested. For a 4% interest rate, the formula is $I = 0.04P$.
Explanation Think of this as a perfect partnership. When one variable changes, the other changes by the exact same multiplier. If you buy twice as many items, you pay twice the price. The ratio between the two quantities always stays constant.
Common Questions
What is direct variation?
Direct variation is a relationship of the form y = kx, where k is the constant of variation (or proportionality). As x increases, y increases proportionally.
How do you find the constant of variation?
Divide any y-value by its corresponding x-value: k = y/x. This ratio is constant for all points in a direct variation.
How is direct variation different from a general linear equation?
Direct variation always passes through the origin (0, 0), while a general linear equation y = mx + b may not pass through the origin when b ≠ 0.
What does the graph of direct variation look like?
The graph of y = kx is a straight line through the origin, with slope k. Positive k gives an upward slope; negative k gives a downward slope.