Problem 60 - Algebra 2

Question

Suppose \(z\) varies jointly with \(x\) and \(y .\) Write a function that models each relationship. Find the value of \(z\) when \(x=-2\) and \(y=3 .\) \(z=-5\) when \(x=-1\) and \(y=-1\)

Step-by-Step Solution

Answer

The value of \(z\) when \(x=-2\) and \(y=3\) is -30.

Step 1 - Set Up the Variation Model

Since \(z\) varies jointly with \(x\) and \(y\), the variation model is in the form \(z = k \cdot x \cdot y\), where \(k\) is the constant of variation.

Step 2 - Solve for the Constant

To find the value of \(k\), substitute one of the given sets of values into the equation and solve for \(k\). Given \(z=-5\) when \(x=-1\) and \(y=-1\), plug in these values to find \(k\). \n\n So we have \(-5=-k(-1)(-1)\). Solving for \(k\), we find that \(k=5\).

Step 3 - Substitute Values to Find \(z\)

Now we have a complete model: \(z = 5 \cdot x \cdot y\). Substitute the second set of given values, \(x=-2\) and \(y=3\), into the equation to find the corresponding value of \(z\). \n\n So, \(z = 5 \cdot -2 \cdot 3\). Solving this gives us \(z = -30\).

Key Concepts

Constant of Variation

In the realm of mathematical relationships, the constant of variation plays a vital role in connecting variables when they are related through a variation. In the context of joint variation, where a variable, say "z", varies with other variables like "x" and "y", the constant of variation is denoted by "k." This constant expresses the proportionality between variables. For instance, if "z" varies jointly with "x" and "y", our equation becomes: \( z = k \cdot x \cdot y \) Here, "k" ensures that changes in "x" and "y" directly influence "z" in a predictable way. Finding the constant of variation involves substituting known values into the variation equation to solve for "k." Let's consider an example: when given that \( z = -5 \) when \( x = -1 \) and \( y = -1 \), you substitute these values to form the equation \( -5 = k \cdot (-1) \cdot (-1) \). Solving this shows that \( k = 5 \). Thus, once "k" is identified, it forms a bridge that holds the relationship between the variables intact.

Modeling Functions

Modeling functions in mathematics is all about creating an equation or a rule that describes the relationship between variables. In joint variation scenarios, this involves equating one variable as a product of others, influenced by a constant. For example, to model the relationship where "z" varies jointly with "x" and "y", we write the function as \( z = k \cdot x \cdot y \). This representation provides a formula that, once the constant of variation "k" is known, lets you compute "z" for any values of "x" and "y". Once you have your model ready, you can predict outcomes and analyze patterns. For example, substituting back into the modeled function with known values allows you to explore scenarios and check consistency. Modeling functions not only assists in solving direct problems but also enhances comprehension of how variables interact with each other in a systematic way. Whether in science, economics, or daily-life scenarios, modeling gives us the power to represent and solve complex problems descriptively and quantitatively.

Algebraic Substitution

Algebraic substitution is a fundamental technique used to solve equations and find unknown values more easily. It involves replacing a variable in an equation with a known value or expression. In our joint variation case, after modeling the function and finding the constant "k", substitution allows us to calculate unknown variables when other values are given. Imagine having the model function: \( z = 5 \cdot x \cdot y \). To find "z" for \( x = -2 \) and \( y = 3 \), you replace "x" and "y" in the equation with these numbers. This leads to: \( z = 5 \cdot (-2) \cdot 3 \), which simplifies to \( z = -30 \). The elegance of algebraic substitution lies in its simplicity and power. By simply replacing variables with numbers or expressions, you transform the equation itself, making it easier to drive toward a solution. This technique is a cornerstone in not just joint variation problems but across a multitude of algebraic inquiries and problem-solving situations.