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Chapter 7: Problem 10

\(x y=\frac{1}{5}\)

Short Answer

Expert verified
The equation can be solved for one variable in terms of the other. For \(y\), the expression is \(y = \frac{1}{5x}\). For \(x\), the expression is \(x = \frac{1}{5y}\).

Step by step solution

01

Identify the Variables

In the given equation \(x y=\frac{1}{5}\), \(x\) and \(y\) are two variables.
02

Set up the Equation for \(y\) in terms of \(x\)

Given that \(x y=\frac{1}{5}\), we can rearrange the equation in terms of \(y\). We can achieve this by dividing both sides of the equation by \(x\). So, we get \(y = \frac{1}{5x}\).
03

Set up the Equation for \(x\) in terms of \(y\)

Similarly, we can also express \(x\) in terms of \(y\). By dividing both sides of the equation by \(y\), we get \(x = \frac{1}{5y}\).

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Key Concepts

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

Understanding Variables
In algebra, variables are symbols that represent unknown values or quantities. They can take the form of letters, like \(x\) and \(y\) in the original equation \(xy = \frac{1}{5}\). Variables serve as placeholders that can represent any number within an equation until they're solved.
For instance, both \(x\) and \(y\) can each have various numerical values and make the equation hold true depending on their relationship. This relationship is described by the equation itself.
  • Variables allow flexibility and generalization in equations.
  • They are crucial for expressing equations and formulas elegantly.
  • Understanding how variables interact is key to solving algebraic problems.
In this context, the equation conveys a particular mathematical relationship between \(x\) and \(y\) through multiplication, which is key to further manipulation.
Equation Rearrangement
Equation rearrangement is an essential algebraic skill. It involves changing the form of an equation without altering its underlying truth. This is often done to isolate particular variables for solving.
Rearranging involves basic operations such as addition, subtraction, multiplication, and division. These operations allow you to manipulate the equation. In our original problem, we rearranged the equation \(xy = \frac{1}{5}\) to express one variable in terms of the other:
  • For \(y\): Dividing both sides by \(x\) gives us \(y = \frac{1}{5x}\).
  • For \(x\): Dividing both sides by \(y\) results in \(x = \frac{1}{5y}\).
By rearranging the equation, we gain insights into the relationship between the variables and can solve for one based on the known value of the other.
Solving for a Variable
Solving for a variable means finding its specific value, often by expressing it in terms of known quantities or other variables. This process relies on the technique of isolating the variable you're interested in.
The equation \(xy = \frac{1}{5}\) can be manipulated to solve for either variable. The steps include:
  • Identifying which variable should be isolated. For example, to solve for \(y\), we start with \(xy = \frac{1}{5}\).
  • Applying the inverse operation to isolate the variable. Here, dividing both sides by \(x\) yields \(y = \frac{1}{5x}\).
  • Ensuring the solution is valid by checking it against the original equation context.
This method not only helps find specific values but also provides a general form of the variable in relation to others in the equation, making it adaptable for different situations or datasets.

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Most popular questions from this chapter

In Exercises 33-40, rewrite the function in the form \(g(x)=\frac{a}{x-h}+k\). Graph the function. Describe the graph of \(g\) as a transformation of the graph of \(f(x)=\frac{a}{x}\). $$ g(x)=\frac{7 x-20}{x+13} $$

You are hired to wash the new cars at a car dealership with two other employees. You take an average of 40 minutes to wash a car \(\left(R_1=1 / 40\right.\) car per minute \()\). The second employee washes a car in \(x\) minutes. The third employee washes a car in \(x+10\) minutes. a. Write expressions for the rates that each employee can wash a car. b. Write a single expression \(R\) for the combined rate of cars washed per minute by the group. c. Evaluate your expression in part (b) when the second employee washes a car in 35 minutes. How many cars per hour does this represent? Explain your reasoning.

Your friend claims it is possible for a rational function to have two vertical asymptotes. Is your friend correct? Justify your answer.

In Exercises 33-40, rewrite the function in the form \(g(x)=\frac{a}{x-h}+k\). Graph the function. Describe the graph of \(g\) as a transformation of the graph of \(f(x)=\frac{a}{x}\). $$ g(x)=\frac{4 x-11}{x-2} $$

Find the least common multiple of the expressions. \(24 x^2, 8 x^2-16 x\)
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