Problem 34 - College Algebra

Question

For the following exercises, use the given information to find the unknown value. \(y\) varies jointly as \(x\) and \(z\). When \(x=4\) and \(z=2\), then \(y=16\). Find \(y\) when \(x=3\) and \(z=3\).

Step-by-Step Solution

Answer

The value of \( y \) is 18 when \( x = 3 \) and \( z = 3 \).

Step 1: Understanding Joint Variation

When we say that a variable \( y \) varies jointly as \( x \) and \( z \), it means that \( y = kxz \) for some constant \( k \). The variable \( y \) depends on the product of \( x \) and \( z \). Our goal is to find this constant \( k \) and use it to calculate the new value of \( y \).

Step 2: Determine the Constant \( k \)

Given that \( y = 16 \) when \( x = 4 \) and \( z = 2 \), substitute these values into the joint variation equation: \[ 16 = k \times 4 \times 2 \]Simplify the right side:\[ 16 = 8k \]Solve for \( k \) by dividing both sides by 8:\[ k = \frac{16}{8} = 2 \]

Step 3: Use the Constant to Find New \( y \)

Now that we have \( k = 2 \), we use this in the joint variation equation for the new values of \( x \) and \( z \):\[ y = 2 \times 3 \times 3 \]Multiply the values together:\[ y = 2 \times 9 \]Simplify:\[ y = 18 \]

Key Concepts

Constant of Variation

The concept of the "Constant of Variation" plays a crucial role in understanding how variables relate to each other in certain algebraic problems. When dealing with joint variation, the constant, usually denoted as \( k \), expresses the proportionality among the variables involved. In joint variation, where one variable \( y \) varies jointly with two other variables \( x \) and \( z \), we express this relationship as:\[ y = kxz \]This equation suggests that \( y \) is directly proportional to the product of \( x \) and \( z \). The constant \( k \) remains the same throughout different scenarios as long as the conditions of the problem are met. In our specific problem, given that \( y = 16 \) when \( x = 4 \) and \( z = 2 \), we find \( k \) with:\[ 16 = k imes 4 imes 2 \]Simplifying this equation allows us to determine that \( k = 2 \). Identifying the constant \( k \) is essential before any other calculations can be made because it establishes the proportional framework for solving the problem.

Product of Variables

The term "Product of Variables" in a joint variation problem involves multiplying two or more variables to determine how they jointly affect another variable. In joint variation problems like the one presented, we see \( y \) varies with the product \( xz \), which can be mathematically expressed as:\[ y = kxz \]Here, the "product of variables" refers to the \( xz \) component of the equation. This equation showcases how \( x \) and \( z \) together influence \( y \) via their multiplication, mediated by the constant \( k \). For instance, if \( x \) and \( z \) each double, \( y \) would quadruple provided \( k \) stays constant. When determining the unknown value of \( y \) for new values of \( x \) and \( z \), simply plug these values into:\[ y = k imes 3 imes 3 \]Here \( y \) will equal the product of \( k \) with \( 9 \), resulting from multiplying the given values of \( x \) and \( z \). Such problems highlight how changes in multiple inputs affect the outcome.

Algebraic Problem Solving

The art of "Algebraic Problem Solving" is illustrated well in exercises involving joint variation. Here, problem solving involves understanding the relationships between variables and using algebraic manipulation to find unknown values. The process typically starts with understanding the variation equation appropriate for the problem, in this case:\[ y = kxz \]Next, by substituting known values into the equation, one can solve for the constant \( k \), which was determined here by solving:\[ 16 = 8k \]After finding \( k \), problem solving continues by substituting the new values of \( x \) and \( z \) back into the equation to solve for the new \( y \). This sequence of steps allows us to determine that \( y = 18 \) when \( x = 3 \) and \( z = 3 \). Algebraic problem solving decomposes complex situations into a series of manageable steps and logical reasoning, resulting in clear and accurate solutions to the problems at hand.