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Chapter 7: Problem 74

Find an equation of variation in which: \(y\) varies jointly as \(x\) and \(z\) and inversely as \(w,\) and \(y=\frac{3}{2}\) when \(x=2, z=3,\) and \(w=4\)

Short Answer

Expert verified
The equation is \( y = \frac{xz}{w} \).

Step by step solution

01

- Understand the problem

Identify that this is a joint and inverse variation problem. The variation statement 'y varies jointly as x and z and inversely as w' translates to an equation of the form: \[ y = k \frac{xz}{w} \] where \( k \) is the constant of proportionality.
02

- Substitute given values to find k

Given values: \( y = \frac{3}{2} \), \( x = 2 \), \( z = 3 \), and \( w = 4 \). Substitute these values into the equation to find \( k \): \[ \frac{3}{2} = k \frac{(2)(3)}{4} \]
03

- Simplify the equation

Simplify the equation to isolate \( k \): \[ \frac{3}{2} = k \frac{6}{4} \] Then, simplify \( \frac{6}{4} \) to \( \frac{3}{2} \): \[ \frac{3}{2} = k \frac{3}{2} \]
04

- Solve for k

Divide both sides by \( \frac{3}{2} \) to isolate \( k \): \[ k = 1 \]
05

- Write the final equation

Substitute \( k \) back into the variation equation: \[ y = k \frac{xz}{w} \] Substituting \( k = 1 \), the final equation is: \[ y = \frac{xz}{w} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

constant of proportionality
In any variation problem, the constant of proportionality plays a key role. It is denoted by the letter \( k \). This constant helps in maintaining the proportional relationship between variables. For example, in the equation \( y = k \frac{xz}{w} \), \( k \) ensures that changes in \( x, z, \) and \( w \) are reflected in \( y \) in a proportional manner.
Think of it as a scaling factor. Once you identify \( k \), you can predict the behavior of the variables under different circumstances.
In this way, the constant of proportionality serves as a bridge, connecting the variables in a meaningful way.
proportional relationships
Proportional relationships describe how the variables interact with one another. When two quantities vary jointly, like \( x \) and \( z \) in this exercise, it means they increase or decrease together. On the other hand, if a quantity varies inversely like \( y \) with \( w \), it means \( y \) will do the opposite of \( w \).
In simpler terms:
  • If \( x \) doubles and everything else stays the same, \( y \) also doubles.
  • If \( w \) doubles and everything else stays the same, \( y \) halves.
The relationship between the variables follows the proportional aspect highlighted by the constant of proportionality, making the problem easier to navigate.
variation equations
Variation equations are mathematical expressions describing how one variable depends on others. This problem involves a mixed variation, specifically joint and inverse variation. The general formula for joint and inverse variation, in this case, is \( y = k \frac{xz}{w} \).
Let's break it down:
  • Joint variation: \( y \) depends directly on \( x \) and \( z \). As \( x \) or \( z \) increases, \( y \) increases, given \( w \) stays the same.
  • Inverse variation: \( y \) depends inversely on \( w \). As \( w \) increases, \( y \) decreases, given both \( x \) and \( z \) are unchanged.
This combined setup allows for more complex relationships between variables, providing deeper insights into their behavior.
algebraic manipulation
Algebraic manipulation refers to the processes and techniques used to solve for unknowns in equations. In this exercise:
  • First, identify the relationship \( y = k \frac{xz}{w} \)
  • Next, substitute the known values into the equation \( \frac{3}{2} = k \frac{(2)(3)}{4} \)
  • Simplify the right-hand side \( \frac{3}{2} = k \frac{6}{4} \)
  • Further simplify to \( \frac{3}{2} = k \frac{3}{2} \)
  • Then isolate \( k \) by dividing both sides by \( \frac{3}{2} \)
  • The result is \( k = 1 \)
  • Finally, substitute \( k \) back into the main equation to get the final equation \( y = \frac{xz}{w} \)
These steps involve combining operations like multiplication, division, and simplification, all integral to algebraic manipulation.

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