/*! 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 26 Find the solution set to each eq... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 26

Find the solution set to each equation. $$\frac{5}{x}=\frac{7}{9}$$

Short Answer

Expert verified
The solution set is \(x = \frac{45}{7}\).

Step by step solution

01

Cross-multiply

To solve the equation \(\frac{5}{x} = \frac{7}{9}\), start by cross-multiplying to eliminate the fractions. Multiply 5 by 9 and 7 by x: \[5 \times 9 = 7 \times x\]
02

Simplify the equation

Simplify the expression obtained in Step 1. You get: \[45 = 7x\]
03

Solve for x

Isolate x by dividing both sides of the equation by 7: \[x = \frac{45}{7}\]

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.

Cross-Multiplication
Cross-multiplication is a method used to solve rational equations, especially those involving fractions. When you have an equation in the form of \(\frac{a}{b} = \frac{c}{d}\), you can eliminate the fractions by cross-multiplying.
This means you multiply the numerator of one fraction by the denominator of the other fraction.
Following this method, for the equation \(\frac{5}{x} = \frac{7}{9}\), you multiply 5 (numerator of the first fraction) by 9 (denominator of the second fraction) and 7 (numerator of the second fraction) by x (denominator of the first fraction):
\[5 \times 9 = 7 \times x\]
This gives you an equation without fractions, making the next steps simpler.
Isolating Variables
Isolating variables is a technique used to solve equations by getting the variable by itself on one side of the equation. After cross-multiplying in the equation \(\frac{5}{x} = \frac{7}{9}\), you will have:
\[45 = 7x\]
To isolate x, you need to divide both sides of the equation by 7. This step is crucial to solve for x:
\[7x = 45\]
\[x = \frac{45}{7}\]
In this way, you have isolated x and found its value. Always perform the same operation on both sides of the equation to maintain equality.
Fractions in Algebra
Fractions in algebra can often make an equation look more complicated.
However, using proper techniques like cross-multiplication can simplify them. When dealing with fractions, it helps to understand concepts such as:
  • Numerators and denominators - these are the top and bottom parts of a fraction, respectively.
  • Cross-multiplication - useful for solving equations with fractions on both sides.
  • Equivalent fractions - understanding that \(\frac{a}{b} = \frac{c}{d}\) implies \a \times d = b \times c\.
In the equation \(\frac{5}{x} = \frac{7}{9}\), recognizing that the fractions can be eliminated through cross-multiplication simplifies the problem significantly.
By converting the fractions to a statement without them, you can easily solve for the variable.

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

Perform the indicated operations. Variables in exponents represent integers. $$\frac{x^{3 a+1}}{y^{2 b-3}} \cdot \frac{y^{3 b+4}}{x^{2 a-1}}$$

Perform the indicated operations. $$\frac{5-10 k}{k^{2}-2 k} \div \frac{2 k^{2}+7 k-4}{k^{2}+2 k-8}$$

Solve each equation. $$\frac{5}{2 w+6}-\frac{1}{w-1}=\frac{1}{w+3}$$

Solve each equation. $$\frac{9}{x^{3}-1}-\frac{1}{x-1}=\frac{2}{x^{2}+x+1}$$

Either solve the given equation or perform the indicated operation \((s),\) whichever is appropriate. $$\frac{4}{x}=\frac{3}{4}$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

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.