How to Write and Solve Direct Variation Equations

Here’s a comprehensive guide on how to write and solve direct variation equations:

1. Understanding Direct Variation

A direct variation can be represented by the equation:

\(y = kx\)

Here, (\(y\)) and (\(x\)) are the variables that vary directly with each other, and (\(k\)) is the constant of variation or the constant of proportionality.

2. Writing Direct Variation Equations

To write a direct variation equation, follow these steps:

1. Identify the Variables:
   Identify the two variables that are in direct variation. Let’s denote them as (\(y\)) and (\(x\)).
2. Find the Constant of Variation:
   Use the given information to find the value of (\(k\) ). This can be done by rearranging the direct variation equation as: \(k = \frac{y}{x}\) Substitute the given values of (\(y\)) and (\(x\)) to find (\(k\)).
3. Write the Equation:
   Once you have the value of (\(k\)), substitute it back into the equation (\(y= kx\)) to write the direct variation equation.

3. Solving Direct Variation Equations

To solve a direct variation equation, follow these steps:

1. Isolate the Variable:
   If you are given a direct variation equation and asked to solve for one variable, rearrange the equation to isolate that variable. For example, to solve for (\(x\)), rearrange the equation as: \(x= \frac{y}{k}\)
2. Substitute the Given Values:
   Substitute the given values of the other variable and the constant of variation into the equation and solve for the unknown variable.

4. Examples

Example 1:

Given that (\(y\)) varies directly with (\(x\)), and (\(y = 15\)) when (\(x = 5\)), write the direct variation equation and find (\(y\)) when (\(x = 10\)).

Solution:

1. Find the Constant of Variation:
   \(k = \frac{y}{x} = \frac{15}{5} = 3\)
2. Write the Direct Variation Equation:
   \(y = 3x\)
3. Find (\(y\)) when (\( x = 10\)):
   \(y = 3 \cdot 10 = 30\)

Example 2:

Given the direct variation equation (\(y = 4x\)), find (\(x\)) when (\(y = 20\)).

Solution:

1. Rearrange the Equation to Solve for (\(x\)):
   \(x = \frac{y}{4}\)
2. Substitute the Given Value of (\(y\)):
   \(x = \frac{20}{4} = 5\)

Direct variation equations represent simple proportional relationships between two variables. By understanding the form of the equation (\(y=kx\)) and how to isolate variables, you can easily write and solve direct variation equations for various problems in mathematics and science.