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: Since y varies inversely as x, we can express this relationship as y = k/x, where k is the constant of variation.
2. Use the given values to find the constant of variation k: Substitute y = 12 and x = 5 into the equation y = k/x to solve for k.
3. Solve for k: Multiply both sides of the equation by x to isolate k, resulting in k = y * x.
4. Substitute the known values into the equation: k = 12 * 5.
5. Use the constant of variation to find the new value of y: Substitute k and the new value of x (x = 2) into the equation y = k/x to find y.

Key Concepts

Inverse Variation

Inverse variation describes a relationship where one variable increases as the other decreases, typically expressed as y = k/x, where k is a constant. In this case, if y varies inversely as x, then the product of x and y remains constant. Understanding this relationship is crucial for solving problems involving inverse variation.

Constant of Variation

The constant of variation (k) is the product of the two variables in an inverse variation scenario. It can be determined by substituting known values of x and y into the equation y = k/x. Once k is found, it can be used to find the value of y for any other given value of x in the same variation problem.

Four-Step Problem-Solving Procedure

The four-step procedure for solving variation problems typically includes: 1) identifying the relationship between the variables, 2) determining the constant of variation, 3) setting up the equation using the constant, and 4) solving for the unknown variable. This structured approach helps ensure that all necessary steps are followed to arrive at the correct solution.