Direct Variation - Algebra-2

1. Fundamental Concepts

Definition: When two variables x and y satisfy $y = kx$ (where k is a non-zero constant), y is said to vary directly with x (direct variation).
Core Feature: The value of one variable is a fixed multiple of the value of the other variable, i.e., output value = input value × constant.
Significance of the Constant: The k in the formula is called the constant of variation, which represents the proportional relationship between the two variables.

2. Key Concepts

Basic Rule: $$y = kx$$

Identifying Direct Variation: If $$\frac{y}{x} = k$$ for all values of $$x \neq 0$$ , then $$y$$ varies directly with $$x$$ .

Application: Direct variation models many real-world phenomena such as speed and distance, pressure and volume in gases, etc.

3. Examples

Easy Level

Determine whether y varies directly with x in the following table. If so, find the constant of variation.

x	1	4	5
y	2	8	10

Solution: y varies directly with x, the constant of variation $k = 2$, and the function expression is $y = 2x$. (Basis: $\frac{2}{1} = \frac{8}{4} = \frac{10}{5} = 2$)

Medium Level

Given that y varies directly with x, and $y = 6$ when $x = -15$. Find the value of y when $x = 20$.

Solution:

Let the function expression be $y = kx$;
Substitute $x = -15$ and $y = 6$, we get $6 = -15k$, so $k = -\frac{2}{5}$;
When $x = 20$, $y = -\frac{2}{5} \times 20 = -8$.

Hard Level

Students in Ms. Lee's class are going on a field trip to a math museum. The total cost varies directly with the number of admission tickets bought. It costs $243 for 18 students. Find the total cost for 27 students.

Solution:

Let the total cost be C and the number of students be t, then $C = kt$;
Substitute $C = 243$ and $t = 18$, we get $243 = 18k$, so $k = 13.5$;
When $t = 27$, $C = 13.5 \times 27 = 364.5$ dollars.

4. Problem-Solving Techniques

Judging Whether It Is a Direct Variation:
- For tabular data: Calculate the ratios of all $\frac{y}{x}$; if they are equal, it is a direct variation.
- For equations: If it can be simplified to the form $y = kx$ (without a constant term), it is a direct variation.
Finding the Constant of Variation k:
- If a set of values $(x, y)$ is known, directly calculate using $k = \frac{y}{x}$.
Solving Practical Problems:
- Steps: ① Set up the direct variation relationship (e.g., $y = kx$); ② Substitute the known data to find k; ③ Use the obtained k to solve the new problem.