Direct and Inverse VARIATION Section 8.1

  a is called the constant of variation. We will do an example together.

If y varies directly as x, and y =24 and x=3 find: (a) the constant of variation (b) Find y when x=2 (a) Find the constant of variation y = ax 24 = a•3 a = 8 Write the general equation Substitute

(b) Find y when x=2 First we find the constant of variation, which was a = 8 Now we substitute into y = ax. y = ax y = 8•2 y = 16

Another method of solving direct variation problems is to use proportions. So lets look at a problem that can by solved by either of these two methods.

If y varies directly as x and y=6 when x=5, then find y when x=15. Proportion Method:

Now lets solve using the equation. y = ax y = ax   Either method gives the correct answer, choose the easiest for you.

Now you do one on your own. y varies directly as x, and x=8 when y=9. Find y when x=12. Answer: 13.5

Inverse Variation   Just as with direct variation, a proportion can be set up solve problems of indirect variation.

Write an inverse variation equation The variables x and y vary inversely and y=7 when x = 4. Write an equation that relates x and y. Then find y when x = -2.          

Find y when x=15, if y varies inversely as x and x=10 when y=12 Solve by equation:    

A general form of the proportion Lets do an example that can be solved by using the equation and the proportion.

Solve by proportion: Find y when x=15, if y varies inversely as x and x=10 when y=12

Try it yourself!! Solve this problem using either method. Find x when y=27, if y varies inversely as x and x=9 when y=45. Answer: 15

Review of Variations Direct Variation Inverse Variation Formula General Equation

Inverse or Direct??    

Hint: Solve the equation for y and take notice of the relationship. Ex: tell whether x & y show direct variation, inverse variation, or neither. xy=4.8 y=x+4 c. Inverse Variation Hint: Solve the equation for y and take notice of the relationship. Neither Direct Variation

Joint Variation When a quantity varies directly as the product of 2 or more other quantities. For example: if z varies jointly with x & y, then z=axy. How would you write the following? If y varies jointly with x and z, then If x varies jointly with y and z, then

Example  

Solve for y If y varies jointly as x and z, and y = 54 when x = 2 and z = 9, find y when x = 7 and z = 10. Since a = 3, then y = axz y = 3•7•10 y = 210

Try it yourself!! If y varies jointly as x and z, and y = 48 when x = 3 and z = 4, find the constant of variation. If x varies jointly as y and z, and x = 36 when y = 36 and z = 4, find x when y = 12 and z = 8. a = 4 x = 24