Joint Variation
Jointly Proportional
When we say z is jointly proportional to a set of variables, it
means that z is directly
proportional to each variable taken one
at a time.
If z varies jointly with respect to x and y,
the equation will be of the form z = kxy (where k is
a constant).
Equation:
c =
5ab
Variable c is jointly proportional
to a and b. That means c is
directly proportional to both a and b.
Doubling a causes c to double. Doubling b causes c to
double. Doubling both a and b causes c to
quadruple.
|
| a |
b |
c |
| 1 |
1 |
5 |
| 2 |
1 |
10 |
| 1 |
2 |
10 |
| 2 |
2 |
20 |
|
See
also
Inverse variation, gravity
Worked Example
Problem: Suppose z varies jointly with x and y. When x = 3 and y = 4, z = 60. Find z when x = 5 and y = 2.
Step 1: Write the joint variation equation.
Step 2: Substitute the known values to find the constant k.
60=k(3)(4)=12k Step 3: Solve for k by dividing both sides by 12.
k=1260=5 Step 4: Write the specific equation with the value of k.
Step 5: Substitute x = 5 and y = 2 to find the new value of z.
z=5(5)(2)=50 Answer: When x = 5 and y = 2, z = 50.
Another Example
This example applies joint variation to a real geometric formula, showing that the well-known triangle area formula is actually an instance of joint variation with k = 1/2.
Problem: The area A of a triangle varies jointly with its base b and height h. A triangle with base 10 cm and height 6 cm has an area of 30 cm². Find the area of a triangle with base 14 cm and height 8 cm.
Step 1: Write the joint variation equation for area.
Step 2: Substitute A = 30, b = 10, and h = 6 to find k.
30=k(10)(6)=60k Step 3: Solve for k.
k=6030=21 Step 4: Use the equation with b = 14 and h = 8.
A=21(14)(8)=56 Answer: The area is 56 cm². Notice that k = 1/2 matches the familiar triangle area formula A = ½bh.
Frequently Asked Questions
What is the difference between joint variation and direct variation?
Direct variation involves one variable being proportional to a single other variable, written as y = kx. Joint variation extends this idea to two or more variables, such as z = kxy. You can think of joint variation as direct variation with multiple factors — z is directly proportional to x and directly proportional to y at the same time.
Can joint variation involve more than two variables?
Yes. Joint variation can involve any number of variables. For example, w varies jointly with x, y, and z means w = kxyz. The key idea stays the same: the dependent variable is proportional to the product of all the independent variables.
How do you find the constant of variation in a joint variation problem?
Substitute one complete set of known values into the equation z = kxy, then solve for k. Once you have k, you can use the equation to find unknown values for any other combination of the variables. Always find k first before answering other parts of the problem.
Joint Variation vs. Inverse Variation
| Joint Variation | Inverse Variation |
|---|
| Definition | z is proportional to the product of two or more variables | y is proportional to the reciprocal of another variable |
| Formula | z = kxy | y = k/x |
| Effect of doubling one variable | The dependent variable doubles | The dependent variable is halved |
| Graph behavior | Linear in each variable when the other is held constant | Hyperbolic curve approaching but never touching the axes |
| Real-world example | Area of a rectangle: A = lw | Travel time at fixed distance: t = d/v |
Why It Matters
Joint variation appears frequently in science and geometry. Newton's law of gravitation, the area formulas for rectangles and triangles, and the ideal gas law are all joint variation relationships. Understanding joint variation helps you model situations where a result depends on multiple factors simultaneously, which is common in physics, engineering, and economics courses.
Common Mistakes
Mistake: Adding the variables instead of multiplying them, writing z = k(x + y) instead of z = kxy.
Correction: 'Varies jointly' means the variables are multiplied together. The word 'jointly' refers to a product, not a sum. Always write z = kxy.
Mistake: Forgetting to find the constant k before solving for the unknown variable.
Correction: You must first use a known set of values to calculate k. Without the correct k, any further computation will be wrong. Substitute the given data point into z = kxy and solve for k as your first step.
