/*! 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 34 \(\frac{x}{x-5}-4=\frac{3}{x-5}\... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 34

\(\frac{x}{x-5}-4=\frac{3}{x-5}\)

Short Answer

Expert verified
The solution is \(x = \frac{17}{3}\).

Step by step solution

01

Identify Common Denominator

Observe that the denominators in the equation are both \(x-5\). This makes it easier to work with because we can use this common denominator to simplify the equation.
02

Simplify the Equation

Since the denominators are the same, we can set the numerators equal to each other after eliminating the denominators: \(x - 4(x-5) = 3\).
03

Distribute and Simplify

Distribute the \(-4\) into \((x-5)\). This gives us: \(x - 4x + 20 = 3\). Simplify it to \(-3x + 20 = 3\).
04

Solve for x

Rearrange the equation \(-3x + 20 = 3\) to find \(x\). First, subtract 20 from both sides to get \(-3x = -17\). Divide both sides by \(-3\) to solve for \(x\): \(x = \frac{17}{3}\).

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.

Common Denominator
A common denominator is a crucial concept when working with fractions, especially in algebra. By finding a common denominator, you can simplify complex equations. In our exercise, the common denominator is already evident: both fractions have the denominator \(x-5\). This makes the problem easier to solve.
  • With a common denominator like \(x-5\), you can eliminate the fractions by focusing on the numerators.
  • This step is essential because it transforms the problem from dealing with fractions to a simpler linear equation.
Recognizing common denominators is a handy skill not just for solving algebraic equations but is also useful when adding or subtracting fractions in any context.
Simplifying Expressions
Simplifying expressions involves combining and reducing terms to make an equation easier to understand and solve. In this exercise, we simplify by eliminating the common denominator and simplifying the resulting expression.
  • First, remove the common denominator. This step turns our equation into one without fractions: \(x - 4 = 3\).
  • Distribute any constants across terms in parentheses. In this case, \(-4\) is distributed across \((x-5)\).
After distributing, the expression \(x - 4x + 20 = 3\) simplifies to \(-3x + 20 = 3\). This simplified expression is critical for solving the equation in the next steps.
Solving Linear Equations
Solving linear equations involves finding the value of the variable that makes the equation true. With our example equation simplified, solving it becomes straightforward.
  • First, isolate terms involving \(x\). In \(-3x + 20 = 3\), you start by getting rid of any constants on the side of \(x\).
  • Subtract 20 from both sides to get \(-3x = -17\).
  • Next, divide both sides by the coefficient of \(x\) to solve for it. Here, divide by \(-3\) to get \(x = \frac{17}{3}\).
Linear equations are foundational in algebra and mastering them helps in understanding more complex algebraic concepts. The ability to break down and solve these step by step is a valuable mathematical skill.

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 multiplications and divisions and express your answers in simplest form. $$\frac{4 x}{11 y} \div \frac{12 x}{33}$$

Add or subtract as indicated and express your answers in simplest form. (Objective 3) $$\frac{3 x}{4}+\frac{x}{6}-\frac{5 x}{8}$$

Perform the indicated multiplications and divisions and express your answers in simplest form. $$\frac{6 a b}{4 a b+4 b^{2}} \div \frac{7 a-7 b}{a^{2}-b^{2}}$$

Add or subtract as indicated and express your answers in simplest form. (Objective 3) $$\frac{3}{x-5}+\frac{7}{x}$$

Perform the indicated operations and express the answers in simplest form. Remember that multiplications and divisions are done in the order that they appear from left to right. $$\frac{5 x y^{2}}{12 y} \cdot \frac{18 x^{2}}{15 y} \div \frac{3}{2 x y}$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Decision 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.