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.