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

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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.

Use the given values to find the constant $k$. Substitute $y = 10$ and $x = 3$ into the equation $y \times x = k$ to get $10 \times 3 = k$.

Calculate the constant $k$ by multiplying the given values: $k = 30$.

Write the inverse variation equation with the constant $k$: $y \times x = 30$.

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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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