Solve each problem. If y varies inversely as x, and y=10 when x=3, find y when x=20.

Verified step by step guidance

1. Identify the relationship: Since y varies inversely as x, we can express this as y = k/x, where k is a constant.
2. Use the given values to find the constant k: Substitute y = 10 and x = 3 into the equation y = k/x to get 10 = k/3.
3. Solve for k: Multiply both sides of the equation by 3 to find k, resulting in k = 10 * 3.
4. Substitute the value of k back into the inverse variation equation: Now that we have k, the equation becomes y = k/x.
5. Find y when x = 20: Substitute x = 20 into the equation y = k/x and use the value of k found earlier to solve for y.

Key Concepts

Inverse Variation

Inverse variation describes a relationship between two variables where one variable increases as the other decreases. Mathematically, this is expressed as y = k/x, where k is a constant. In this case, if y varies inversely as x, it means that the product of x and y remains constant for all values of x and y.

Finding the Constant of Variation

To solve problems involving inverse variation, the first step is to determine the constant of variation (k). This is done by substituting the known values of x and y into the equation y = k/x. For example, if y = 10 when x = 3, we can find k by rearranging the equation to k = y * x, resulting in k = 10 * 3 = 30.

Solving for Unknowns

Once the constant of variation is known, we can find the value of y for any given x by substituting x into the inverse variation equation. For instance, if we want to find y when x = 20, we use the equation y = k/x with k = 30. Thus, y = 30/20, which simplifies to y = 1.5, allowing us to solve for the unknown variable.