U36 Variations: Essential Formulas

This section covers the fundamental concepts of variation, including direct, inverse, joint, and partial variation, which are crucial for solving DSE problems involving relationships between variables.

1 Direct Variation

Definition and Formula

A variable $y$ is said to vary directly as (or be directly proportional to) another variable $x$ if there exists a non-zero constant $k$ such that $y = kx$. The constant $k$ is called the constant of variation.

$$ y \propto x \quad \Rightarrow \quad y = kx $$
$x$$y$$y = kx$ (k>0)$

2 Inverse Variation

Definition and Formula

A variable $y$ varies inversely as (or is inversely proportional to) $x$ if there exists a non-zero constant $k$ such that $y = \frac{k}{x}$, or equivalently, $xy = k$.

$$ y \propto \frac{1}{x} \quad \Rightarrow \quad y = \frac{k}{x} $$
$x$$y$$y = \frac{k}{x}$ (k>0)$

3 Joint Variation

Definition and Formula

A variable $z$ varies jointly as $x$ and $y$ if $z$ is directly proportional to the product of $x$ and $y$. That is, $z = kxy$ for some non-zero constant $k$.

$$ z \propto x y \quad \Rightarrow \quad z = kxy $$

4 Partial Variation

Definition and Formula

A variable $y$ is said to vary partly as $x$ and partly as a constant. This gives a linear relationship of the form $y = kx + c$, where $k$ and $c$ are constants with $k \neq 0$.

$$ y = kx + c $$

5 Combined Variation

General Form

Many real-world problems involve a combination of direct and inverse variations. The general form is $z \propto \frac{x^m}{y^n}$, which translates to $z = k \frac{x^m}{y^n}$, where $k$ is the constant of variation and $m$, $n$ are real numbers.

$$ z = k \frac{x^m}{y^n} $$

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