Lesson 8: Finding Direct Variation

New Concept

When a problem states that $A$ varies directly as $B$, the equation $A = kB$ is implied. This relationship is called direct variation.

Why it matters

Direct variation is your first tool for translating real-world patterns into the language of algebra. Mastering this simple linear relationship is the foundation for modeling everything from chemical reactions to economic trends.

What’s next

Next, you’ll translate word problems into the equation $A=kB$, find the constant of variation, and solve for unknown values.

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!

The constant $k$ in the equation $A = kB$ is called the constant of variation.

1. A scooter travels 10 km in 30 minutes ($D=kT$). To find k, set up $10 = k(30)$, so the constant is $k = \frac{1}{3}$. 2. If 120 seconds have passed in 2 minutes ($S=kM$), then $120 = k(2)$, so the constant of variation is $k = 60$.

Meet 'k', the secret code of the relationship! This number is the specific multiplier that never changes for a given situation. Find 'k' first, and you can solve any problem related to that variation.

The statement $A$ varies directly as $B$ also implies the proportion:

1. If 12 items cost 78 dollars, how much for 42 items? $\frac{78}{C_2} = \frac{12}{42} \rightarrow 78 \cdot 42 = 12 \cdot C_2 \rightarrow C_2 = 273$ dollars. 2. If 15 ties are for 30 feet of track, how many for 50 feet? $\frac{15}{T_2} = \frac{30}{50} \rightarrow 15 \cdot 50 = 30 \cdot T_2 \rightarrow T_2 = 25$ ties.

This is a fantastic shortcut for solving problems when you don't need to find the constant 'k'. Just set up the two pairs of values as a proportion, cross-multiply, and you've found your answer!