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Chapter 4: Problem 37

Solve the equation and simplify your answer. $$x-\frac{4}{7}=\frac{7}{8}$$

Short Answer

Expert verified
The solution to the equation is \(x = \frac{81}{56}\).

Step by step solution

01

Understand the Equation

We need to solve the equation \(x - \frac{4}{7} = \frac{7}{8}\). Our goal is to find the value of \(x\) that makes this equation true.
02

Isolate the Variable

To solve for \(x\), we add \(\frac{4}{7}\) to both sides of the equation in order to isolate \(x\). This gives us \(x = \frac{7}{8} + \frac{4}{7}\).
03

Find a Common Denominator

The denominators 8 and 7 need a common denominator to add the fractions. The smallest common denominator is 56. Now, convert each fraction: \(\frac{7}{8} = \frac{49}{56}\) and \(\frac{4}{7} = \frac{32}{56}\).
04

Add the Fractions

Add the fractions: \(\frac{49}{56} + \frac{32}{56} = \frac{81}{56}\).
05

Simplify the Fraction (if necessary)

The fraction \(\frac{81}{56}\) is already in its simplest form since 81 and 56 do not have any common factors other than 1.

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Key Concepts

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

Understanding Fractions
Fractions are numerical expressions that represent a part of a whole. A fraction consists of two numbers: the numerator and the denominator. The numerator, located above the line, shows how many parts we have, while the denominator, below the line, shows how many parts make up a whole. Let's look at an example, \( \frac{4}{7} \). Here, 4 is the numerator and 7 is the denominator.
  • If the numerator is smaller than the denominator, the fraction is less than one.
  • A numerator larger than the denominator indicates a fraction greater than one, also known as an improper fraction.
  • If the numerator is equal to the denominator, the fraction is equivalent to one.
When solving equations with fractions, like in our problem, it's important to understand how to manipulate them.
Finding a Common Denominator
When adding or subtracting fractions, finding a common denominator is essential. This is because you can only directly add or subtract fractions when they share the same denominator. To find a common denominator:
  • Identify the denominators you need to work with.
  • Find the smallest number that both denominators can divide into without leaving a remainder, which is the least common multiple (LCM).
  • Convert the original fractions to equivalent fractions with this common denominator.
Let's apply this to our exercise: we're dealing with denominators of 8 and 7. The smallest common multiple of 8 and 7 is 56. This means we convert:\( \frac{7}{8} = \frac{49}{56} \) and \( \frac{4}{7} = \frac{32}{56} \).With common denominators, the fractions can easily be added or subtracted.
Simplifying Fractions
Simplifying fractions makes them easier to work with and understand. A fraction is in its simplest form when the greatest common divisor (GCD) of both the numerator and the denominator is 1. This means there are no numbers other than 1 that can divide both the numerator and the denominator.
  • Check if both the numerator and the denominator share any common factors.
  • If they do, divide both by their GCD to reduce the fraction.
In our exercise, after adding \( \frac{49}{56} + \frac{32}{56} = \frac{81}{56} \),we check for a common factor between 81 and 56. Since their GCD is 1, the fraction \( \frac{81}{56} \)is already in its simplest form. Understanding how to simplify helps keep computations cleaner and results clearer.

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Most popular questions from this chapter

Solve the equation and simplify your answer. $$-\frac{3}{2} x=-\frac{5}{9}$$

Is \(-8 / 15\) a solution of the equation \(\frac{1}{4} x=-\frac{1}{15}\) ?

Solve the equation and simplify your answer. $$\frac{1}{2} x-\frac{1}{2}=\frac{3}{2} x+\frac{3}{4}$$

Solve the equation and simplify your answer. $$\frac{1}{6} x=\frac{2}{9}$$

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