Problem 16 - College Algebra

Question

Write an equation that expresses each relationship. Then solve the equation for y. \(x\) varies jointly as \(y\) and \(z\) and inversely as the square of \(w\)

Step-by-Step Solution

Answer

Based on the given relationship, the equation is \(x = k \cdot \frac{{y \cdot z}}{{w^2}}\). When solved for \(y\), it is \(y = \frac{{x \cdot w^2}}{{k \cdot z}}\)

Step 1: Write the equation

Based on the given relationship, we can write the equation as \(x = k \cdot \frac{{y \cdot z}}{{w^2}}\), where \(k\) is the constant of variation.

Step 2: Solve the equation for y

We now isolate \(y\) in the equation. Multiply both sides by \(w^2\) and then divide by \(z\), we get: \(y = \frac{{x \cdot w^2}}{{k \cdot z}}\)

Key Concepts

Joint Variation

Joint variation describes a relationship where a variable changes proportionally with the product of two or more other variables. For example, if a variable x varies jointly with variables y and z, it implies that as y and z increase or decrease, x also increases or decreases in direct proportion to their product. The general formula for joint variation is x = k * y * z, where k is the constant of variation, which remains fixed as y and z change.

In the context of our exercise, the joint variation between x, y, and z is expressed as part of a larger equation involving both joint and inverse variation. Joint variation allows us to understand how two or more factors can work together to influence another variable in a direct and proportional manner.

Inverse Variation

Inverse variation describes a relationship where a variable changes in the opposite direction and in proportion to the inverse of another variable. In simple terms, as one variable increases, the other decreases, and vice versa. The relationship is characterized by the formula x = k / w, where x directly varies with the inverse of w, and k is the constant of variation.

In our exercise, the variable x varies inversely as the square of w, which is noted as x being proportional to 1/w^2. This signifies that an increase in w will result in a decrease in x, and when w decreases, x will increase, provided the value of k remains constant. Understanding inverse variation is crucial for determining the ebb and flow of relationships where variables move out of sync but in a predictable pattern.

Solving for y

Solving for y in an equation with multiple variables involves isolating y on one side of the equation. To do this, we perform operations that 'undo' what is currently being applied to y. In this case, since y is being multiplied by z and divided by w², we need to reverse these operations.

We multiply the entire equation by w² to eliminate the division by w², and then we divide by z to cancel out the multiplication by z. This yields the solution y = (x * w²) / (k * z). By carefully performing operations that preserve the equality, we're able to solve for one variable in terms of the others, facilitating easier understanding and manipulation of the given relationship.

Constant of Variation

The constant of variation, denoted by k, is a fixed value that relates the variables in an equation of direct or inverse variation. The constant remains unchanged as the variables it relates vary. It is essentially the 'glue' that holds the relationship together.

In the equation x = k * (y * z) / w², k must be determined from known values of x, y, z, and w in order to use the equation for further predictions or calculations. Once found, the constant of variation allows us to predict the behavior of one variable when the others change, as long as the relationship described by the equation remains true. In our step-by-step solution, k is factored out when solving for y, and it is vital for the equation to represent the relationship accurately.