/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 51 Solve each system by the method ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 51

Solve each system by the method of your choice. $$ \left\\{\begin{array}{l} {\frac{3}{x^{2}}+\frac{1}{y^{2}}=7} \\ {\frac{5}{x^{2}}-\frac{2}{y^{2}}=-3} \end{array}\right. $$

Short Answer

Expert verified
The solutions to the system of equations are \( (\sqrt{2}, \frac{1}{2\sqrt{2}}) \), \( (\sqrt{2}, -\frac{1}{2\sqrt{2}}) \), \( (-\sqrt{2}, \frac{1}{2\sqrt{2}}) \), and \( (-\sqrt{2}, -\frac{1}{2\sqrt{2}}) \).

Step by step solution

01

Add the two equations

First, add the two equations together. Doing this will cause the terms containing \( \frac{1}{y^{2}} \) to cancel each other out. This will leave you with an equation containing only terms of \( \frac{1}{x^{2}} \), which is easier to handle. Here's what you get: \( \frac{3}{x^{2}} + \frac{1}{y^{2}} + \frac{5}{x^{2}} - \frac{2}{y^{2}} = 7 - 3 \). After doing the addition and subtraction, you get \( \frac{8}{x^{2}} = 4 \).
02

Solve for x

Next, solve for \( x \) in the equation \( \frac{8}{x^{2}} = 4 \). Multiply both sides by \( x^{2} \) to get rid of the denominator on the left side. This gives \( 8 = 4x^{2} \). Then divide each side by 4 to isolate \( x^{2} \). You should get \( x^{2} = 2 \). Finally, take square roots of both sides to solve for x, yielding \( x = \sqrt{2} \) and \( x = -\sqrt{2} \).
03

Substitute \( x \) into one of the original equations

To find the value of \( y \), substitute \( x = \sqrt{2} \) and \( x = -\sqrt{2} \) into one of the original equations. In this case, we can choose the first equation, \( \frac{3}{x^{2}} + \frac{1}{y^{2}} = 7 \). Substituting \( x = \sqrt{2} \) gives \( \frac{3}{2} + \frac{1}{y^{2}} = 7 \). Solving this equation for \( y^{2} \), we get \( y^{2} = \frac{1}{8} \) and thus \( y = \frac{1}{2\sqrt{2}} \) and \( y = -\frac{1}{2\sqrt{2}} \). Doing the same for \( x = -\sqrt{2} \) yields the same solutions for \( y \). Thus, our solutions are \( (\sqrt{2}, \frac{1}{2\sqrt{2}}) \), \( (\sqrt{2}, -\frac{1}{2\sqrt{2}}) \), \( (-\sqrt{2}, \frac{1}{2\sqrt{2}}) \), and \( (-\sqrt{2}, -\frac{1}{2\sqrt{2}}) \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Substitution Method
The substitution method is a powerful technique for solving systems of equations where one variable is expressed in terms of another. It's particularly helpful when dealing with equations that involve complex fractions or irrational numbers. Here's how it generally works:
  • First, solve one of the equations for one variable in terms of the other. This makes it easier to handle and isolate each variable separately.
  • Next, substitute this expression into the other equation. This allows you to eliminate one of the variables, creating a single equation with only one variable to solve.
  • Solve the simplified equation for the remaining variable.
  • Once you have the value of one variable, substitute it back into either of the original equations to find the other variable.
In the original exercise, though we directly added the equations, the concept of substitution can also be applied. Solve one of the rational equations for either \( x^2 \) or \( y^2 \), and substitute back to resolve the system.
Solving Rational Equations
Rational equations feature fractions where the numerators and/or denominators contain variables. Solving these types of equations often requires finding common denominators or simplifying complex fractions. To solve rational equations successfully:
  • First, eliminate the fractions. This can be done by multiplying every term by the least common denominator (LCD) of all fractions present. This helps to clear the denominators.
  • Aim to simplify the resulting equation to make solving it easier. This might involve factoring or combining like terms.
  • After terms are simplified, solve the equation like a standard equation, isolating the variable of interest.
  • Finally, verify your solutions in the original equation. Due to the removal of denominators, always ensure no solutions produce a zero in a denominator.
In our problem, simplifying the equations led us to deal with just a single term in the variable, making it straightforward to solve.
Square Roots
Square roots are the inverse operation of squaring a number and are essential in solving equations like \( x^2 = a \). Knowing how to handle square roots is crucial when solving equations that arise from squaring.When you encounter a square in an equation, to solve for that variable:
  • First, isolate the squared term on one side of the equation. This prepares it for the square root process.
  • Take the square root of both sides of the equation. This operation has two possible solutions: the positive and negative root.
  • Calculate these roots to find the potential values of the variable.
  • Always check these solutions back in the original equation, as squaring can introduce extraneous solutions.
In the exercise, solving \( x^2 = 2 \) required taking the square roots, which introduced potential solutions of \( x = \sqrt{2} \) and \( x = -\sqrt{2} \). Applying the same operation to \( y^2 \) confirmed the solution set.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

An object moves in simple harmonic motion described by \(d=6 \cos \frac{3 \pi}{2} t,\) where \(t\) is measured in seconds and \(d\) in inches. Find: a. the maximum displacement b. the frequency c. the time required for one cycle. (Section \(5.8, \text { Example } 8)\)

You invest in a new play. The cost includes an overhead of \(\$ 30,000,\) plus production costs of \(\$ 2500\) per performance. A sold-out performance brings in \(\$ 3125 .\) (In solving this exercise, let \(x\) represent the number of sold-out performances.

Harsh, mandatory minimum sentences for drug offenses account for more than half the population in U.S. federal prisons. The bar graph shows the number of inmates in federal prisons, in thousands, for drug offenses and all other crimes in 1998 and 2010. (Other crimes include murder, robbery, fraud, burglary, weapons offenses, immigration offenses, racketeering, and perjury.) a. In 1998, there were 60 thousand inmates in federal prisons for drug offenses. For the period shown by the graph, this number increased by approximately 2.8 thousand inmates per year. Write a function that models the number of inmates, y, in thousands, for drug offenses x years after 1998. b. In 1998, there were 44 thousand inmates in federal prisons for all crimes other than drug offenses. For the period shown by the graph, this number increased by approximately 3.8 thousand inmates per year. Write a function that models the number of inmates, y, in thousands, for all crimes other than drug offenses x years after 1998. c. Use the models from parts (a) and (b) to determine in which year the number of federal inmates for drug offenses was the same as the number of federal inmates for all other crimes. How many inmates were there for drug offenses and for all other crimes in that year?

\(g(x)=\frac{x-6}{x^{2}-36}\) (Section 3.5, Example 1)

Solve: \(x^{4}+2 x^{3}-x^{2}-4 x-2=0\) (Section \(3.4, \text { Example } 5)\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.