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 that if y varies inversely as x, it means their product is constant. This can be written as the equation \(y \times x = k\), where \(k\) is a constant.
2. Use the given values \(y = 10\) when \(x = 3\) to find the constant \(k\). Substitute these values into the equation: \$10 \times 3 = k$.
3. Calculate the value of \(k\) from the equation \(k = 10 \times 3\) (do not compute the final number, just set up the expression).
4. Now, use the constant \(k\) and the new value \(x = 20\) to find the corresponding \(y\). Substitute into the inverse variation formula: \(y \times 20 = k\).
5. Solve for \(y\) by isolating it on one side: \(y = \frac{k}{20}\). Substitute the expression for \(k\) from step 3 to express \(y\) in terms of known values.

Key Concepts

Inverse Variation

Inverse variation describes a relationship where one variable increases as the other decreases, following the rule y = k/x. Here, k is a constant, meaning the product of x and y remains the same for all values.

Finding the Constant of Variation

To solve inverse variation problems, first find the constant k by multiplying the given x and y values. This constant helps determine unknown values by maintaining the equation k = x * y.

Solving for an Unknown Variable

Once the constant k is known, substitute the new x value into the equation y = k/x to find the corresponding y. This step applies the inverse variation formula to find missing values.