Use the four-step procedure for solving variation problems given on page 447 to solve Exercises 1–10. y varies directly as x. y = 65 when x = 5. Find y when x = 12.

Verified step by step guidance

1. Identify the type of variation described. Since y varies directly as x, we can write the equation as \(y = kx\), where \(k\) is the constant of proportionality.
2. Use the given values to find the constant \(k\). Substitute \(y = 65\) and \(x = 5\) into the equation \(y = kx\) to get \(65 = k \times 5\).
3. Solve for \(k\) by dividing both sides of the equation by 5, resulting in \(k = \frac{65}{5}\).
4. Write the specific variation equation using the value of \(k\) found: \(y = kx\) becomes \(y = \left(\frac{65}{5}\right) x\).
5. Find \(y\) when \(x = 12\) by substituting \(x = 12\) into the equation \(y = \left(\frac{65}{5}\right) x\) and simplify to express \(y\) in terms of known values.

Key Concepts

Direct Variation

Direct variation describes a relationship where one variable is a constant multiple of another, expressed as y = kx. Here, y changes proportionally with x, meaning if x doubles, y also doubles. Understanding this helps set up the equation to find unknown values.

Constant of Variation

The constant of variation (k) is the fixed multiplier linking x and y in direct variation. It is found by substituting known values of x and y into y = kx. Once k is determined, it can be used to find y for any given x.

Four-Step Procedure for Variation Problems

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