Problem 17 - College Algebra

Question

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies jointly as \(x, z,\) and \(w\) and when \(x=1, z=2,\) \(w=5,\) then \(y=100\).

Step-by-Step Solution

Answer

The equation is \( y = 10xzw \).

Step 1: Define the Joint Variation Equation

When a variable varies jointly with two or more variables, we multiply those variables together and introduce a constant of variation. In this case, since \( y \) varies jointly as \( x, z, \text{ and } w \), we have the equation: \( y = kxzw \).

Step 2: Substitute Known Values to Find the Constant

We use the values given in the problem to find the constant \( k \). Substitute \( x = 1 \), \( z = 2 \), \( w = 5 \), and \( y = 100 \) into the equation: \( 100 = k \times 1 \times 2 \times 5 \).

Step 3: Solve for the Constant of Variation

Calculate the product of the values on the right-hand side: \( 1 \times 2 \times 5 = 10 \). Now we have: \( 100 = 10k \). Solve for \( k \) by dividing both sides by 10: \( k = \frac{100}{10} = 10 \).

Step 4: Write the Final Equation

Now that we have found \( k = 10 \), we can write the final equation that describes the joint variation: \( y = 10xzw \).

Key Concepts

Constant of Variation

In mathematics, particularly in algebra, the constant of variation plays a crucial role in describing relationships between variables. It is the number that relates how one variable changes with respect to others in an equation. When we talk about "joint variation," this constant becomes significant because it scales the effect that each of the variables has on the outcome.
In the given example, the relationship between the variables is expressed as a joint variation:

• This means that the variable \( y \) changes in proportion to the product of other variables \( x, z, \text{ and } w \).
• To establish this relation, a constant of variation \( k \) is introduced in the equation \( y = kxzw \).
• This constant \( k \) acts like a bridge, linking the change in \( y \) with changes in \( x, z, \text{ and } w \).
• Determining the constant of variation allows us to predict values of \( y \) for different combinations of \( x, z, \text{ and } w \).

It's a powerful concept that encapsulates complex relationships in a simple form, helping us to make accurate calculations and predictions.

Algebraic Equations

An algebraic equation is a mathematical statement that shows the equality of two expressions. In the context of joint variation, the equations help us formalize how changes in multiple factors affect another variable. Understanding algebraic equations is crucial because:

• They provide a clear and concise way of expressing mathematical relationships.
• Equations like \( y = kxzw \) allow us to capture the essence of joint variation, outlining how variables are interdependent.
• Solving these equations involves substituting known values to find unknowns, such as the constant of variation.

In practice:

• You first identify the form of the equation based on the type of variation (joint, direct, inverse, etc.). Here, it's joint, so the form is \( y = kxzw \).
• Next, known values are substituted into the equation, and algebraic manipulation is used to solve for the unknown constant \( k \).

The elegance of algebraic equations is in their ability to transform abstract concepts into tangible numbers, revealing insights about the relationships they describe.

Problem Solving Steps

Solving problems in algebra, such as finding the equation of joint variation, involves a series of systematic steps. By following these steps, you can solve problems more efficiently and accurately. Consider these problem-solving steps:

• Define the Equation: Start by setting up the right type of equation. For joint variation, you know the form \( y = kxzw \).
• Substitute Known Values: Insert the values provided by the problem to replace the variables in the equation. This common step helps ground the abstract equation in concrete numbers.
• Solve for the Unknown: Simplifying the equation leads to the calculation of missing components, like the constant \( k \). Algebraic techniques, such as multiplication, division, or factoring, can be used here.
• State the Final Equation: Once \( k \) is known, write the specific equation that describes the relationship outlined in the problem. This conclusion highlights the result in a clear format, for example, \( y = 10xzw \).

These problem-solving steps make tackling even complex equations manageable and ensure that each piece is understood before moving on to the next stage.