These lessons help Algebra students learn about joint or combined variation.
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Joint Variation or Combined Variation is when one quantity varies directly as the product of at least two other quantities.
For example:
y = kxz
y varies jointly as x and z, when there is some nonzero constant k
Joint Variation Examples
Example:
Suppose y varies jointly as x and z. What is y when x = 2 and z = 3, if y = 20 when x = 4 and z = 3?
Example:
z varies jointly with x and y.
When x = 3, y = 8, z = 6. Find z, when x = 6 and y = 4.
Joint Variation Application
Example:
The energy that an item possesses due to its motion is called kinetic energy. The kinetic energy of an object (which is measured in joules) varies jointly with the mass of the object and the square of its velocity.
If the kinetic energy of a 3 kg ball traveling 12 m/s is 216 Joules, how is the mass of a ball that generates 250 Joules of energy when traveling at 10 m/s?
Distinguish between Direct, Inverse and Joint Variation
Example:
Determine whether the data in the table is an example of direct, inverse or joint variation. Then, identify the equation that represents the relationship.
In Algebra, sometimes we have functions that vary in more than one element. When this happens, we say that the functions have joint variation or combined variation. Joint variation is direct variation to more than one variable (for example, d = (r)(t)). With combined variation, we have both direct variation and indirect variation.
How to set up and solve combined variation problems?
Example:
Suppose y varies jointly with x and z. When y = 20, x = 6 and z = 10. Find y when x = 8 and z =15.
Lesson on combining direct and inverse or joint and inverse variation
Example:
y varies directly as x and inversely as the square of z, and when x = 32, y = 6 and z = 4. Find x when y = 10 and z = 3.
How to solve problems involving joint and combined variation?
Examples:
Combined Variation
Example:
y varies jointly as x and z and inversely as w, and y = 3/2, when x = 2, z =3 and w = 4. Find the equation of variation.
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