Joint Variation

High School

Definition

A variable z that varies jointly with x and y means that you will be dealing with an equation in the form of z = kxy, where k stands for a constant. Sometimes, this is also referred to as jointly proportional. A example of this would be A = π\(r^2\), which is the equation for the area of a circle.

Worked examples

\(z = 12xy\)

Here \(z\) varies jointly with \(x\) and \(y\), and the constant of variation \(k = 12\).

\(z\) varies jointly with \(x\) and \(y\). If \(z = 24\) when \(x = 2\) and \(y = 3\), then \(24 = k(2)(3)\), so \(k = 4\).

Use the given values to solve for the constant \(k\), then write \(z = 4xy\).

\(A = \pi r^2\)

The area varies jointly with \(r\) and \(r\) (i.e., \(r^2\)), where \(k = \pi\).

Common mistakes

\(z = k + xy\) → \(z = kxy\) Joint variation is multiplication, not addition. The constant multiplies the product of the variables.
\(z = kx + ky\) → \(z = kxy\) Jointly means \(z\) varies with the product \(xy\), not the sum \(x + y\).
If \(z = 6\) when \(x = 2, y = 3\), then \(k = 6\) → \(6 = k(2)(3)\) so \(k = 1\) Substitute all three values into \(z = kxy\) and solve for \(k\); don't just copy \(z\).

Where you'll use it next

Joint variation appears in physics formulas (kinetic energy varies jointly with mass and the square of velocity), geometry (volume and surface-area formulas), and later in multivariable calculus and modeling real-world relationships.

Found in 1 StudyPug lesson

Joint and Combined Variation: Formulas, Examples, and Applications

College Algebra

Unlock the power of joint and combined variation! Learn essential formulas, explore real-world applications, and master problem-solving techniques to excel in advanced algebra and beyond.

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