Problem 18 - College Algebra

Question

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

Step-by-Step Solution

Answer

The equation that expresses the given relationship is \(x = kzu\). After substituting for \(u = y - w\) and solving for \(y\), the final equation is \(y = \frac{x+kwz}{kz}\).

Step 1: Formulate the Joint Variation Equation

For a joint variation relationship, we have a general equation, \(k = \frac{x}{zy}\), where k is the constant of variation. In this case, however, \(x\) varies jointly as \(z\) and the difference between \(y\) and \(w\) (let's call this u, so \(u=y-w\)). Hence the equation is \(k = \frac{x}{zu}\), or equivalently, \(x = kzu\).

Step 2: Substitute for \(u\)

Substitute \(u = y - w\) into the equation. Now we have \(x = kz(y - w)\).

Step 3: Solving for \(y\)

To solve for \(y\), we should isolate \(y\) on one side of the equation. Distribute k and z on the right side to get \(x = kzy - kwz\). Next, rearrange the equation by adding \(kwz\) to both sides which yields \(x + kwz = kzy\). To isolate \(y\), divide both sides by \(kz\), giving us \(y = \frac{x+kwz}{kz}\).

Key Concepts

Understanding the Constant of Variation

In mathematics, understanding the constant of variation is crucial when dealing with joint variation problems. The constant of variation, denoted by \( k \), is a fixed number that relates variables in a direct variation relationship. It remains the same regardless of the specific values of the variables involved.
In the context of joint variation, which concerns multiple variables, we see this concept in action. For instance, when \( x \) varies jointly with \( z \) and another expression such as \( y-w \), we form the equation \( x = k(z(y-w)) \). Here, \( k \) acts as the constant that ties these quantities together.
Identifying this constant is essential because it helps in composing the initial equation. Once you set up the equation correctly with the constant, solving the rest becomes a straightforward algebraic process.

Solve Equations by Substitution

Solving equations, particularly in joint variation problems, often involves substitution. Substitution is a powerful technique that simplifies the equation by expressing one variable in terms of others.
In the provided problem, the expression \( y-w \) can be termed as \( u \) for brevity. We begin by representing the equation as \( x = kzu \). This is a relatively simplified form.
Afterwards, substitute back to express \( u = y-w \), resulting in the equation \( x = kz(y-w) \). This substitution helps in keeping track of variable relationships more easily, turning a potentially complex problem into something manageable. This is particularly useful when combining variables and constants in joint variations.

Isolate Variables to Solve for Them

In algebra, isolating variables is a fundamental skill used to solve equations and find specific variable values. This technique revolves around getting the variable you want to solve for on one side of the equation, making it the subject of the formula.
In the original problem, you aim to solve for \( y \). Start with the equation \( x = kz(y - w) \). Distributing \( kz \) results in \( x = kzy - kwz \).
To isolate \( y \), you first rearrange the equation: add \( kwz \) to both sides, yielding \( x + kwz = kzy \).
Finally, divide every term by \( kz \) to isolate \( y \), resulting in the simplified form \( y = \frac{x + kwz}{kz} \). This step-by-step approach to isolating variables not only simplifies solving equations but also strengthens overall problem-solving skills in mathematical contexts.