History

Variation models grew from early physics and astronomy, where scientists needed formulas for how one quantity depends on another. The language of proportionality became a bridge between algebra and real measurements.

Derivation and reasoning

Why we use a constant \(k\)

“Directly proportional” means ratio \(\frac{y}{x}\) stays constant, so \(\frac{y}{x}=k\) and \(y=kx\). For inverse variation, product \(xy\) stays constant, so \(xy=k\).

Checkpoints

If \(y\propto x\) and \(y=15\) when \(x=3\), find \(y\) when \(x=10\).

Answer: \(k=5\Rightarrow y=5x\Rightarrow y=50\).

If \(y\propto \frac1x\) and \(y=8\) when \(x=5\), find \(y\) when \(x=20\).

Answer: \(k=40\Rightarrow y=\frac{40}{x}\Rightarrow y=2\).

\(y\) varies as \(x^2\). If \(y=27\) when \(x=3\), find \(y\) when \(x=5\).

Answer: \(y=kx^2,\ k=3\Rightarrow y=3(25)=75\).

\(y\) varies jointly as \(x\) and \(z\). If \(y=30\) when \(x=3,z=2\), find \(y\) when \(x=5,z=4\).

Answer: \(y=kxz,\ k=5\Rightarrow y=5(5)(4)=100\).

A taxi fare \(F\) varies partly as distance \(d\): \(F=md+c\). If \(F=26\) at \(d=8\), and \(F=38\) at \(d=14\), find \(m,c\).

Answer: Solve two equations: \(8m+c=26,\ 14m+c=38\).

\(6m=12\Rightarrow m=2,\ c=10\). So \(F=2d+10\).

Applications

• Physics: pressure, speed, and force models often use direct/inverse variation.
• Business: cost models combine fixed and variable components.
• Engineering: scaling laws use power variation relations.

References

• Standard high-school algebra units on direct and inverse variation.
• Elementary modeling texts for joint and partial variation.