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. Understand the concept of inverse variation: If y varies inversely as x, it means that the product of y and x is a constant. Mathematically, this is expressed as \(y \times x = k\), where \(k\) is a constant.
2. Use the given values to find the constant \(k\). Substitute \(y = 10\) and \(x = 3\) into the equation \(y \times x = k\) to get \$10 \times 3 = k$.
3. Calculate the constant \(k\) by multiplying the given values: \(k = 30\).
4. Write the inverse variation equation with the constant \(k\): \(y \times x = 30\).
5. Find \(y\) when \(x = 20\) by substituting \(x = 20\) into the equation and solving for \(y\): \(y \times 20 = 30\), then solve for \(y\) by dividing both sides by 20.

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.

Finding the Constant of Variation

To solve inverse variation problems, first find the constant k by multiplying the given values of x and y. Using y = k/x, substitute the known values to calculate k, which remains the same for all pairs of x and y.

Solving for Unknown Variable

After determining the constant k, substitute the new value of x into the equation y = k/x to find the corresponding y. This step applies the inverse variation formula to find unknown values based on the constant.