/*! 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 12 Solve each equation. $$\frac{x... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 12

Solve each equation. $$\frac{x}{3}+\frac{2 x}{5}=\frac{-11}{5}$$

Short Answer

Expert verified
The solution to the equation is \( x = -3 \).

Step by step solution

01

Identify the goal and the equation

We aim to solve the equation \( \frac{x}{3} + \frac{2x}{5} = \frac{-11}{5} \). This means finding the value of \( x \) that makes the equation true.
02

Find a common denominator

Identify a common denominator for the fractions. The common denominator between 3 and 5 is 15. Rewrite each term with this common denominator.
03

Rewrite the fractions with the common denominator

Convert \( \frac{x}{3} \) to \( \frac{5x}{15} \) and \( \frac{2x}{5} \) to \( \frac{6x}{15} \). The equation now becomes \( \frac{5x}{15} + \frac{6x}{15} = \frac{-33}{15} \).
04

Combine like terms

Add the fractions on the left-hand side since they have the same denominator: \( \frac{5x + 6x}{15} = \frac{11x}{15} \). The equation now is \( \frac{11x}{15} = \frac{-33}{15} \).
05

Clear the fractions

Multiply both sides of the equation by 15 to eliminate the denominators: \( 11x = -33 \).
06

Solve for x

Divide both sides by 11 to isolate \( x \): \( x = -3 \).

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
When dealing with fractions in an equation, finding a common denominator is crucial. This is because adding or subtracting fractions with different denominators is not straightforward. To add fractions readily:
  • Identify the denominators of the fractions involved.
  • Find the smallest multiple both denominators share, which is known as the least common denominator (LCD).
In this problem, the denominators are 3 and 5. The smallest number that both 3 and 5 divide into evenly is 15, which is our common denominator. Using a common denominator allows us to rewrite fractions so they are compatible for addition or subtraction. This step prepares us to combine fractions seamlessly.
Fractions
Fractions consist of a numerator (the top number) and a denominator (the bottom number). They represent parts of a whole. In equations, they often require special attention to be manipulated correctly.When rewriting fractions with a new common denominator, you must:
  • Multiply both the numerator and the denominator by the same number so that the new denominator matches the common denominator.
For instance, to modify \( \frac{x}{3} \) with the common denominator of 15:
  • Multiply numerator (x) and denominator (3) by 5, getting \( \frac{5x}{15} \).
This process ensures the value of the fraction remains the same while the denominator matches others in the equation. Proper handling of fractions allows for easier manipulation and solves the equation effectively.
Combining Like Terms
Combining like terms is a fundamental step in simplifying equations. It involves merging terms with the same variables and powers.In this context:
  • After standardizing the fractions' denominators, fractions like \( \frac{5x}{15} \) and \( \frac{6x}{15} \) become simpler to add.
You can then combine them by adding the numerators, as they share a denominator:
  • The sum is \( \frac{5x + 6x}{15} = \frac{11x}{15} \).
Combining like terms makes the equation less complex and further sets the stage for solving the variable. This process highlights the importance of simplification in algebraic operations.
Isolation of Variable
The ultimate goal in solving a linear equation is to isolate the variable, making it the sole entity on one side of the equation.Here's how you achieve this:
  • First, eliminate fractions by multiplying both sides of the equation by the common denominator.
This transforms \( \frac{11x}{15} = \frac{-33}{15} \) to \( 11x = -33 \).
  • Next, divide each side by the coefficient of the variable to isolate it. For this equation, divide by 11.
This yields:
  • \( x = -3 \)
The isolation of the variable simplifies the equation to its essential form, providing the solution directly. This approach ensures clarity and precision, enabling precise solutions in algebra.

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

The set of real numbers satisfying the given inequality is one or more intervals on the number line. Show the interval(s) on a number line. $$|x|<4$$

Graph the equation after determining the \(x\) - and \(y\) -intercepts and whether the graph possesses any of the three types of symmetry described on page 58 $$y=2 x^{2}$$

Explain why there are no real numbers that satisfy the equation \(\left|x^{2}+4 x\right|=-12\).

(a) Verify that the point (3,7) is on the circle $$x^{2}+y^{2}-2 x-6 y-10=0$$ (b) Find the equation of the line tangent to this circle at the point \((3,7) .\) Hint: A result from elementary geometry says that the tangent to a circle is perpendicular to the radius drawn to the point of contact.

In each of parts (a) through (d), first solve the equation for \(y\) so that you can enter it in your graphing utility. Then use the graphing utility to graph the equation in an appropriate viewing rectangle. In each case, the graph is a line. Given that the \(x\) - and \(y\) -intercepts are (in every case here) integers, read their values off the screen and write them down for easy reference when you get to part (e). (a) \(\frac{x}{2}+\frac{y}{3}=1\) (c) \(\frac{x}{6}+\frac{y}{5}=1\) (b) \(\frac{x}{-2}+\frac{y}{-3}=1\) (d) \(\frac{x}{-6}+\frac{y}{-5}=1\) (e) On the basis of your results in parts (a) through (d), describe, in general, the graph of the equation \(\frac{x}{a}+\frac{y}{b}=1,\) where \(a\) and \(b\) are nonzero constants.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Calculus

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.