Use the four-step procedure for solving variation problems given on page 447 to solve Exercises 1–10. y varies inversely as x. y = 12 when x = 5. Find y when x = 2.

Verified step by step guidance

1. Identify the type of variation described. Since y varies inversely as x, we use the inverse variation formula: \(y = \frac{k}{x}\), where \(k\) is the constant of variation.
2. Use the given values to find the constant \(k\). Substitute \(y = 12\) and \(x = 5\) into the formula: \(12 = \frac{k}{5}\).
3. Solve for \(k\) by multiplying both sides of the equation by 5: \(k = 12 \times 5\).
4. Write the equation with the found constant \(k\): \(y = \frac{k}{x}\), replacing \(k\) with the value found in the previous step.
5. Find \(y\) when \(x = 2\) by substituting \(x = 2\) into the equation: \(y = \frac{k}{2}\). This will give the value of \(y\) for \(x = 2\).

Key Concepts

Inverse Variation

Inverse variation describes a relationship where one variable increases as the other decreases, such that their product is constant. Mathematically, y varies inversely as x means y = k/x, where k is a constant. Understanding this helps set up the equation to find unknown values.

Determining the Constant of Variation

The constant of variation (k) is found by substituting known values of x and y into the variation equation. For inverse variation, k = x * y. This constant remains the same for all pairs of x and y in the problem, allowing calculation of unknown values.

Four-Step Procedure for Variation Problems

This procedure involves: 1) identifying the type of variation, 2) writing the variation equation, 3) finding the constant using given values, and 4) using the constant to find the unknown variable. Following these steps ensures a systematic approach to solving variation problems.