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Direct variation is a fundamental concept in algebra where one variable increases or decreases in direct proportion to another. In simpler words, if one variable goes up, the other goes up at the same rate, and if one goes down, the other does proportionately. In mathematical terms, direct variation can be expressed with the equation: \( y = kx \) where \( y \) varies directly as \( x \) and \( k \) is the constant of proportionality. This constant is the factor that relates the two variables. For example, if \( k = 2 \) and \( x = 3 \) then \( y \) would be \( 6 \) as per the principle of direct variation.

Inverse variation describes a situation where one variable increases as the other decreases, but this time, the product of the two variables remains constant. This is often expressed by the equation \( y = \frac{k}{x} \) or \( xy = k \) where \( k \) is again the constant of variation. If one of the variables is doubled, the other one is halved to maintain the same product value. In the context of our exercise, \( y \) varies inversely as \( x \) means that as \( x \) becomes larger, \( y \) will decrease such that their product equals the constant \( k \).

Algebraic equations are the bread and butter of solving math problems. They consist of symbols and numbers that show the relationship between quantities. Equations are balanced, which means the expressions on both sides are equal in value. When tackling an algebraic equation, the goal is often to find the value of the unknowns – usually represented by letters such as \( x \) or \( y \) – that make the equation true. Equations can be simple, such as linear equations with a single variable, or more complex, involving multiple variables and operations, such as the equation used to express joint variation in the given exercise.

The process of finding the value of variables is what 'solving for variables' entails. The methodology often includes a series of algebraic manipulations like addition, subtraction, multiplication, division, and sometimes more advanced operations such as taking roots or applying exponents. The key to solving for a particular variable is to isolate it on one side of the equation. For example, if dealing with the equation \( ax + b = c \) and you need to solve for \( x \) you would subtract \( b \) from both sides and then divide by \( a \) to get \( x = \frac{c-b}{a} \). The step-by-step solution illustrated how to isolate \( y \) given a joint variation, which involved both multiplication to get rid of a denominator and division to isolate the variable \( y \) on one side of the equation.