Problem 11 - College Algebra

Question

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

Step-by-Step Solution

Answer

The equation expressing the joint variation relationship is \(x = kyz\). After isolating \(y\), the result is \(y = \frac{x}{kz}\).

Step 1: Write the joint variation equation

Because \(x\) varies jointly as \(y\) and \(z\), the joint variation equation is simply \(x = kyz\), where \(k\) is the constant of variation.

Step 2: Isolate y

To isolate \(y\) in the equation \(x = kyz\), it is needed to divide both sides of the equation by \(kz\). This gives the result \(y = \frac{x}{kz}\).

Key Concepts

Algebraic Equations

In the realm of mathematics, understanding algebraic equations is crucial. An algebraic equation represents a relationship between variables using mathematical operations. These can include addition, subtraction, multiplication, division, and exponents. Equations are statements that show two expressions are equal, typically with a variable, such as 'y', to be solved for. It's like a balance scale; whatever you do on one side, you must do on the other to maintain balance.

In our example of joint variation, the algebraic equation is \( x = kyz \). This equation connects three variables with a constant, showing how one variable's change can impact the others. To solve for a specific variable, algebraic principles allow you to manipulate the equation by performing operations to both sides until you isolate the variable of interest.

Constant of Variation

The constant of variation, often represented as \( k \), plays a pivotal role in joint variation equations. It is the unchanging number that relates the variables to one another. Think of it as the glue holding the relationship together. This constant remains the same as long as the relationship between the variables doesn't change.

In joint variation, \( k \) helps maintain the proportionality between \( x \) and the product of \( y \) and \( z \), as seen in the equation \( x = kyz \). Determining the constant's value usually involves having a set of known variables, and then solving the equation for \( k \), giving a complete picture of the relationship.

Solving for Variables

Solving for variables is a fundamental skill in algebra, enabling you to find the value of an unknown in an equation. The process involves isolating the variable on one side of the equation, making it the subject of the formula. The objective is to get the variable by itself, usually through operations that ensure equality is maintained.

When solving for \( y \) in a joint variation equation like \( x = kyz \), you rearrange the equation to express \( y \) in terms of the other variables: \( y = \frac{x}{kz} \). Here, you divide both sides by \( kz \), which leaves \( y \) on its own. The ability to manipulate these equations opens up a deeper understanding of the relationships between quantities and is an essential tool in a variety of scientific and mathematical applications.