If X and Y are in direct proportion then for some non zero real constant k

Y = kx, if X and Y are in direct proportion then for some non zero real constant k

If X and Y are in direct proportion, it means that as X increases (or decreases), Y also increases (or decreases) in a consistent manner. Mathematically, this can be represented as:

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Y = kX

where k is a non-zero constant of proportionality.

This equation indicates that Y is directly proportional to X, with k being the constant of proportionality. This constant signifies the ratio between corresponding values of X and Y. If you multiply X by k, you get Y, and vice versa.

What is Direct Proportion?

Direct proportion, also known as direct variation, is a mathematical relationship between two quantities where increasing one quantity leads to a proportional increase in the other, and vice versa. In simpler terms, they change together in the same direction and at the same rate.

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Here are some key points about direct proportion:

Constant Ratio: The ratio of the two quantities remains constant even when their values change. This constant value is known as the constant of proportionality.
Symbol: Direct proportion is denoted by the symbol "∝". So, if x and y are directly proportional, we write it as x ∝ y.
Formula: Mathematically, direct proportion can be expressed as x = ky, where k is the constant of proportionality. This formula simply states that x is equal to some constant k multiplied by y.
Graphical Representation: When plotted on a graph, two directly proportional quantities form a straight line passing through the origin (0,0).
Real-life Examples: Many real-life situations involve direct proportions. For example:
- The distance traveled by a car is directly proportional to the time it takes for the journey (assuming constant speed).
- The amount of money you pay for groceries is directly proportional to the weight of the groceries you buy (assuming a fixed price per kilogram).
- The number of hours you work is directly proportional to the amount of money you earn (assuming a fixed hourly wage).