Use the four-step procedure for solving variation problems given on page 447 to solve Exercises 1–10.y varies jointly as x and z. y = 25 when x = 2 and z = 5. Find y when x = 8 and z = 12.

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Step 1: Understand the problem. The problem states that 'y varies jointly as x and z'. This means that y is directly proportional to both x and z. In mathematical terms, this can be written as y = kxz, where k is the constant of variation.

Step 2: Use the given values to find the constant of variation. We know that y = 25 when x = 2 and z = 5. Substitute these values into the equation to solve for k: 25 = k * 2 * 5.

Step 3: Once you have found the value of k, you can use it to find y when x = 8 and z = 12. Substitute these values into the equation y = kxz.

Step 4: Simplify the equation to find the value of y.

Step 5: Check your answer by substituting the values of x, y, and z back into the original equation to ensure it holds true.

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Key Concepts

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

Joint Variation

Joint variation occurs when a variable depends on two or more other variables. In this case, y varies jointly as x and z, meaning that y is proportional to the product of x and z. The relationship can be expressed mathematically as y = kxz, where k is a constant of variation.

Finding the Constant of Variation

To solve joint variation problems, the first step is to determine the constant of variation (k). This is done by substituting the known values of y, x, and z into the joint variation equation. For example, using y = 25, x = 2, and z = 5, we can solve for k to establish the relationship between the variables.

Substituting Values

Once the constant of variation is found, the next step is to substitute the new values of x and z into the equation to find the corresponding value of y. This involves using the established equation y = kxz with the new values to calculate y, demonstrating how changes in x and z affect y in joint variation.

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