3.9Direct, Inverse, and Joint Variation
Write direct variation (y = kx), inverse variation (y = k/x), joint variation (y = kxz), and combined variation equations. Find the constant of variation and solve applied problems.
Variation models describe how quantities scale with each other — from Ohm's Law (V = IR) to gravitational force (F ∝ 1/r²). These proportional relationships appear throughout physics, chemistry, and economics.
Essential Question
How can we use variation equations to model and predict real-world relationships between quantities that scale proportionally with each other?
Lesson Overview
Many real-world relationships can be described by variation equations — compact formulas that capture how one quantity changes relative to another. Direct variation (y = kx) describes quantities that increase together; inverse variation (y = k/x) describes quantities where one increases as the other decreases; joint variation (y = kxz) involves two independent variables; and combined variation mixes these types. The constant k, called the constant of variation or constant of proportionality, is found by substituting known values. Mastering variation models is essential for physics, chemistry, engineering, and economics.
Types of Variation
- Direct: y = kx (y ∝ x)
- Inverse: y = k/x (y ∝ 1/x)
- Joint: y = kxz (y ∝ xz)
- Combined: mix of direct and inverse
- Power: y = kxⁿ (direct with a power)
Learning Targets
- Identify the type of variation from a description
- Write the variation equation
- Find the constant of variation k
- Solve for an unknown using the variation model
- Apply variation to real-world contexts
Variation Types
Direct, inverse, and joint
Finding k
4-step process
Real-World Examples
Physics, chemistry, geometry
Worked Examples
y varies directly as x. If y = 18 when x = 6, find y when x = 10.
Step 1 — Write the direct variation equation: y = kx
Step 2 — Substitute known values: 18 = k(6)
Step 3 — Solve for k: k = 18 ÷ 6 = 3
Step 4 — Write the specific equation: y = 3x
Step 5 — Substitute x = 10: y = 3(10) = 30
y varies inversely as x. If y = 8 when x = 3, find y when x = 12.
Step 1 — Write the inverse variation equation: y = k/x
Step 2 — Substitute known values: 8 = k/3
Step 3 — Solve for k: k = 8 × 3 = 24
Step 4 — Write the specific equation: y = 24/x
Step 5 — Substitute x = 12: y = 24/12 = 2
y varies jointly as x and z. If y = 30 when x = 2 and z = 5, find y when x = 4 and z = 3.
Step 1 — Write the joint variation equation: y = kxz
Step 2 — Substitute known values: 30 = k(2)(5) = 10k
Step 3 — Solve for k: k = 30 ÷ 10 = 3
Step 4 — Write the specific equation: y = 3xz
Step 5 — Substitute x = 4, z = 3: y = 3(4)(3) = 36
y varies directly as x² and inversely as z. If y = 12 when x = 2 and z = 1, find y when x = 3 and z = 3.
Step 1 — Write the combined variation equation: y = kx²/z
Step 2 — Substitute known values: 12 = k(2²)/1 = 4k
Step 3 — Solve for k: k = 12 ÷ 4 = 3
Step 4 — Write the specific equation: y = 3x²/z
Step 5 — Substitute x = 3, z = 3: y = 3(9)/3 = 9
The distance d a spring stretches varies directly as the force F applied. A force of 20 N stretches the spring 8 cm. How far will a force of 35 N stretch it?
Step 1 — Write the direct variation equation: d = kF
Step 2 — Substitute known values: 8 = k(20)
Step 3 — Solve for k: k = 8/20 = 0.4 cm/N
Step 4 — Write the specific equation: d = 0.4F
Step 5 — Substitute F = 35: d = 0.4(35) = 14 cm
Guided Practice
y varies directly as x. If y = 21 when x = 7, find y when x = 9.
Hint: Write y = kx, substitute y = 21 and x = 7 to find k, then use y = kx with x = 9.
y varies inversely as x. If y = 5 when x = 4, find y when x = 10.
Hint: Write y = k/x, substitute to find k = 20, then evaluate y = 20/10.
y varies jointly as x and z. If y = 60 when x = 3 and z = 4, find k.
Hint: Substitute into y = kxz: 60 = k(3)(4). Solve for k.
The time t to travel a fixed distance varies inversely as the speed s. If t = 4 hours when s = 60 mph, find t when s = 80 mph.
Hint: Write t = k/s. Find k using t = 4, s = 60. Then substitute s = 80.
y varies directly as x³. If y = 54 when x = 3, find y when x = 2.
Hint: Write y = kx³. Substitute to find k = 2. Then compute y = 2(2³).
Key Vocabulary
Direct Variation
A relationship where y = kx; y is directly proportional to x. As x increases, y increases at a constant rate.
Example: y = 3x — doubling x doubles y
Inverse Variation
A relationship where y = k/x; y is inversely proportional to x. As x increases, y decreases.
Example: y = 12/x — doubling x halves y
Joint Variation
A relationship where y = kxz; y varies directly as the product of two or more variables.
Example: y = 2xz — y doubles when either x or z doubles
Combined Variation
A relationship that involves both direct and inverse variation in the same equation.
Example: y = kx²/z — direct with x², inverse with z
Constant of Variation (k)
The nonzero constant in a variation equation. Found by substituting known values and solving.
Example: If y = 4 when x = 2 in y = kx, then k = 2
Power Variation
A direct variation where y = kxⁿ; y varies directly as a power of x.
Example: A = πr² — area varies directly as r² (k = π)
Check Your Understanding
Interactive Practice — 5 Questions
y varies directly as x. If y = 24 when x = 8, what is the constant of variation k?
y varies inversely as x. If y = 6 when x = 4, find y when x = 8.
Which equation represents "y varies jointly as x and z"?
The area A of a circle varies directly as the square of its radius r. If A = 28.27 when r = 3, which equation models this?
y varies directly as x³. If y = 16 when x = 2, find y when x = 3.
Independent Practice
Independent Practice
y varies directly as x. If y = 35 when x = 5, find y when x = 8.
y varies inversely as x. If y = 9 when x = 4, find y when x = 6.
y varies jointly as x and z. If y = 48 when x = 4 and z = 3, find y when x = 2 and z = 5.
y varies directly as x² and inversely as z. If y = 20 when x = 2 and z = 2, find y when x = 3 and z = 3.
The weight W of an object varies directly as its mass m. If W = 49 N when m = 5 kg, find W when m = 12 kg.
Common Mistakes
Writing y = kx + b for direct variation (confusing with slope-intercept form).
Direct variation is y = kx — the line must pass through the origin. There is no b term.
For inverse variation y = k/x, finding k by dividing y by x instead of multiplying.
From y = k/x, multiply both sides by x to get k = xy. Always isolate k correctly.
In joint variation y = kxz, forgetting to multiply all variables together when finding k.
Substitute all known values into y = kxz and solve: k = y ÷ (xz).
Confusing "varies directly as x²" with "varies directly as x" — using y = kx instead of y = kx².
Read the problem carefully. "Varies directly as x²" means y = kx², not y = kx.
Math Tips
For direct variation y = kx, the graph is always a straight line through the origin. The slope equals k.
For inverse variation y = k/x, the product xy is always constant and equals k. Use this to check your work.
In combined variation, write the equation first, then substitute all known values at once to find k in one step.
Real-world variation problems: identify the type from key words — "directly proportional," "inversely proportional," "varies jointly as."
Always verify your answer by substituting back into the original variation equation with the original known values.