direct variation
Analyze direct variation relationships of the form y=kx: verify that y divided by x is constant, find k, and use it to solve proportional reasoning problems in Grade 10 algebra.
Key Concepts
When the statement of a problem says that $A$ varies directly as $B$ or that $A$ is directly proportional to $B$, the equation $A = kB$ is implied.
1. If pay ($P$) varies directly with hours worked ($H$), the relationship is $P = kH$. If you earn 50 dollars for 5 hours, you'll earn 100 dollars for 10 hours. 2. The resistance ($R$) is directly proportional to the length ($L$), so $R=kL$. A longer wire means proportionally higher resistance.
Think of this as a perfect partnership! When one value changes, the other changes by the exact same multiplier. If you buy more pizza slices, the cost goes up predictably. It’s a straight line relationship!
Common Questions
What is direct variation?
Direct variation is a linear relationship expressed as y=kx, where k is the nonzero constant of variation. As x increases, y increases proportionally. The graph is always a line through the origin with slope k.
How do you write a direct variation equation given one data point?
Substitute the known (x,y) values into y=kx and solve for k. Then write the equation with that k value. For example, if y=15 when x=3, then 15=k*3 gives k=5, and the direct variation equation is y=5x.
How is direct variation used to solve for unknown values?
Once k is known, substitute any new x value to find the corresponding y value, or substitute a known y to find x. For example, with y=5x, if x=7 then y=35. If y=40 then x=8. This proportional reasoning applies to real-world rate and unit-price problems.