Problem 16 - College Algebra with Corequisite Support

Question

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies jointly with \(x\) and \(z\) and when \(x=2\) and \(z=3, \quad y=36\).

Step-by-Step Solution

Answer

The equation is \(y = 6xz\).

Step 1: Understand the Concept

When a variable is said to vary jointly with two other variables, it means that it is directly proportional to the product of the other two variables. In this problem, it means \(y = kxz\), where \(k\) is a constant of proportionality.

Step 2: Define the Equation

The equation for joint variation is given by \(y = kxz\). This implies that \(y\) varies as the product of \(x\) and \(z\) multiplied by the constant \(k\).

Step 3: Substitute Given Values and Solve for k

Substitute the given values into the equation to find \(k\). We know \(x = 2, z = 3\), and \(y = 36\). Substitute these into \(y = kxz\):\[36 = k \cdot 2 \cdot 3\].

Step 4: Calculate the Constant k

We simplify the equation from Step 3: \(36 = 6k\). Solve for \(k\) by dividing both sides by 6:\[k = \frac{36}{6} = 6\].

Step 5: Write the Final Equation

Now that we know \(k\), substitute it back into the joint variation equation:\(y = 6xz\). This is the equation that describes the relationship of the variables.

Key Concepts

Direct Proportionality

When we talk about direct proportionality, we mean that as one variable increases, the other variable also increases at a constant rate. This is expressed in algebraic terms for two quantities, say \(y\) and \(x\), as \(y = kx\), where \(k\) is a constant. In the context of joint variation, like in our exercise, if \(y\) varies jointly with \(x\) and \(z\), then \(y\) is directly proportional to the product of \(x\) and \(z\), giving us the equation \(y = kxz\). So, whenever we deal with direct proportionality in joint variation, remember that we are dealing with the straightforward relationship between variables where they change consistently relative to each other.

Constant of Proportionality

The constant of proportionality is a key component in understanding proportional relationships. It is the factor that relates two variables that are directly proportional. In our example, the constant \(k\) in \(y = kxz\) tells us how much \(y\) will increase with every unit increase of \(x\times z\). To determine \(k\), we use the given values from the problem. We know that when \(x=2\) and \(z=3\), then \(y=36\). By solving \(36 = k \times 2 \times 3\), we find \(k = 6\). This means for every single product of \(x\) and \(z\), \(y\) will be 6 times that value. The constant \(k\) helps us maintain the relationship between the variables consistent and predictable.

Algebraic Equation

An algebraic equation represents a mathematical statement where two expressions are set equal to each other. In our context, we formulate an algebraic equation to describe how \(y\) varies with \(x\) and \(z\) together. The process involves expressing the joint relationship with \(y = kxz\), where all variables are clearly defined. This equation is the backbone of figuring out the nature of their relationship. When solving such an equation, especially in joint variation problems, it's about rewriting the problem statement into a mathematical form that can be interpreted and solved rationally. Once we substitute known values and find the constant \(k\), we get a specific equation \(y = 6xz\) as a clear representation of the variable situation.

Variable Relationship

Understanding the relationship between variables is crucial in solving joint variation problems. Here, \(y\) does not just vary with either \(x\) or \(z\) alone, but with the combination of both. This means that the way \(y\) changes depends on both \(x\) and \(z\) together. So when \(x\) or \(z\) changes, \(y\) responds to the overall effect of both variables. This joint relationship is expressed mathematically by \(y = kxz\). Analyzing this along with given values allows us to pinpoint how exactly each variable contributes to the outcome. Essentially, understanding these relationships empowers us to navigate through values effectively and grasp how changing one affects the others in the particular scenario of joint variation.