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Direct variation is when two variables are related in such a way that the ratio of their values always remains constant. In other words, one variable is a constant multiple of the other. The equation representing direct variation is usually written as follows:
\( y=kx \)
where:

• \( y \) is the dependent variable,
• \( x \) is the independent variable,
• \( k \) is the constant of proportionality.

To find the value of \( k \), we can use given values for \( x \) and \( y \), and solve for \( k \) by rearranging the equation. For example, if \( y = 3 \) and \( x = 2 \), then \( k = \frac{y}{x} = \frac{3}{2} = 1.5 \).
This constant \( k \) stays the same for any pair of values, making it clear that the relationship between \( x \) and \( y \) is consistent and proportional.

Inverse variation describes a situation where the product of two variables remains constant; as one variable increases, the other decreases. The general form of the equation representing inverse variation is:
\( y = \frac{k}{x} \)
where:

• \( y \) is the dependent variable,
• \( x \) is the independent variable,
• \( k \) is the constant of proportionality.

To determine the constant of proportionality \( k \), you use known values for \( x \) and \( y \), and then solve for \( k \). For instance, if \( y = 3 \) when \( x = 2 \), then \( k = y \times x = 3 \times 2 = 6 \). This shows that for any value of \( x \), multiplying it by the corresponding value of \( y \) will always give 6 in this scenario. Variations can also involve different forms such as \( y = \frac{k}{x^2} \) or \( y = \frac{k}{\sqrt{x}} \), where you would adjust the formula to fit these situations appropriately.

Solving equations involves finding the value of the variable that makes the equation true. This can be achieved through various methods like substitution, elimination, or using algebraic manipulations.
For the given problems, solving equations largely involves isolating the variable \( k \). Here’s a step-by-step approach using substitution:

• Identify the relationship and the given values.
• Substitute the known values into the equation.
• Rearrange the equation to solve for the unknown variable.

Take, for example, the equation \( y = \frac{k}{x} \), where you need to find \( k \). If given \( y = 3 \) and \( x = 2 \), you substitute these values in and solve for \( k \): \( 3 = \frac{k}{2} \). Multiply both sides by 2 to get \( k = 6 \).
This method ensures clear and systematic solving of equations, helping you always arrive at the correct value for the unknown variable.