Write and solve direct variation equations

Key Notes:

Direct Variation

  • A direct variation is a relationship between two quantities where one quantity varies directly with the other.
  • In other words, if one quantity increases, the other quantity also increases at a constant rate, and vice versa.

Equation of Direct Variation

  • A direct variation equation can be written in the form y = kx, where:
    • y is the dependent variable
    • x is the independent variable
    • k is the constant of variation

Finding the Constant of Variation

  • To find the constant of variation, divide the dependent variable by the independent variable for any pair of corresponding values.

Solving Direct Variation Equations

  1. Substitute the known values: Substitute the known values for the variables into the direct variation equation.
  2. Solve for the unknown variable: Use algebraic methods to solve for the unknown variable.

Example:

If y varies directly with x, and y = 12 when x = 4, find the equation of variation and the value of y when x = 7.

  1. Find the constant of variation: k = y/x = 12/4 = 3
  2. Write the equation of variation: y = 3x
  3. Find y when x = 7: y = 3(7) = 21

Key Points to Remember:

  • A direct variation is a relationship where one quantity varies directly with the other.
  • The equation of direct variation is y = kx, where k is the constant of variation.
  • To find the constant of variation, divide the dependent variable by the independent variable.
  • To solve a direct variation equation, substitute the known values and solve for the unknown variable.

Learn with an example

Assume that y varies directly with x.If y = 18 when x = 3, find y when x =2.

Write and solve a direct variation equation to find the answer.

y=______

First find the constant of variation. Plug x = 3 and y =18 into the direct variation equation and then solve for k.

y = kx
18 = k(3)
6 = k

Now use k=6 to write the direct variation equation.

y = kx
y = 6x

The direct variation equation is y = 6x.

Finally, use the direct variation equation to find y when x=2.

y = 6x
y = 6(2)
y = 12

so,

y =12.

If y varies directly with x and y = 20 when x = 4, find y when x =2.

Write and solve a direct variation equation to find the answer.

y=________

First find the constant of variation. Plug x = 4 and y =20 into the direct variation equation and then solve for k.

y = kx
20 = k(4)
5 = k

Now use k=5 to write the direct variation equation.

y = kx
y = 5x

The direct variation equation is y = 5x.

Finally, use the direct variation equation to find y when x =2.

y = 5x
y = 5(2)
y = 10

so,

y =10.

If y varies directly with x and y = 20 when x = 10, find y when x=4.

Write and solve a direct variation equation to find the answer.

y=__________

First find the constant o fvariation. Plug x = 10 and y =20 into the direct variation equation and then solve for k.

y = kx
20 = k(10)
2 = k

Now use k=2 to write the direct variation equation.

y = kx
y = 2x

The direct variation equation is y = 2x.

Finally, use the direct variation equation to find y when x=4.

y = 2x
y = 2(4)
y = 8

so

y = 8.

let’s practice!