# How to Write and Solve Direct Variation Equations

Direct variation equations are a fundamental concept in algebra, representing relationships where one variable changes directly in proportion to another.

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:

**Identify the Variables:**

Identify the two variables that are in direct variation. Let’s denote them as (\(y\)) and (\(x\)).**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 (\(k\)).**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:

**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: \(x= \frac{y}{k}\)**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:**

**Find the Constant of Variation:**

\(k = \frac{y}{x} = \frac{15}{5} = 3\)**Write the Direct Variation Equation:**

\(y = 3x\)**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:**

**Rearrange the Equation to Solve for (\(x\)):**

\(x = \frac{y}{4}\)**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.

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