/*! 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 664 In the following exercises, solv... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 664

In the following exercises, solve the equation. $$ x-\frac{1}{6}=\frac{7}{6} $$

Short Answer

Expert verified
x = \( \frac{4}{3} \)

Step by step solution

01

- Understand the equation

The equation given is \( x - \frac{1}{6} = \frac{7}{6} \). We need to solve for \(x\).
02

- Isolate the variable

To solve for \(x\), we need to isolate it on one side of the equation. We can do this by adding \( \frac{1}{6} \) to both sides of the equation.
03

- Perform the addition

Add \( \frac{1}{6} \) to both sides: \( x - \frac{1}{6} + \frac{1}{6} = \frac{7}{6} + \frac{1}{6} \).
04

- Simplify

On the left side: \( x \). On the right side: \( \frac{7}{6} + \frac{1}{6} = \frac{8}{6} \). Simplify \( \frac{8}{6} \) to \( \frac{4}{3} \). Therefore, the equation simplifies to \( x = \frac{4}{3} \).

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Key Concepts

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

Isolating Variables
When solving linear equations, one of the key steps is isolating the variable. This means getting the variable by itself on one side of the equation. In the given exercise, we are solving for \( x \) in the equation \( x - \frac{1}{6} = \frac{7}{6} \).
To isolate \( x \), you can follow these steps:
  • Identify what is being added or subtracted from the variable. Here, \( \frac{1}{6} \) is subtracted from \( x \).
  • Perform the opposite operation on both sides of the equation. Since \( \frac{1}{6} \) is subtracted, we add \( \frac{1}{6} \) to both sides:

$$ x - \frac{1}{6} + \frac{1}{6} = \frac{7}{6} + \frac{1}{6} $$
This simplifies as:
$$ x = \frac{8}{6} $$
Now, \( x \) is isolated on one side of the equation.
Adding Fractions
Adding fractions is a crucial skill in algebra. When fractions have the same denominator, you simply add the numerators. This is what we did in our exercise:
  • We had two fractions with the same denominator: \( \frac{1}{6} \) and \( \frac{7}{6} \).
  • We added the numerators while keeping the denominator the same:

$$ \frac{7}{6} + \frac{1}{6} = \frac{7+1}{6} = \frac{8}{6} $$
This step ensures the fractions are combined correctly. If the denominators are different, you need to find a common denominator before adding the fractions, which might require additional steps.
Simplifying Fractions
Once we have added the fractions, the next step is to simplify the resulting fraction, if possible. Simplifying fractions involves finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by that number.
  • In our exercise, we ended up with the fraction \( \frac{8}{6} \).
  • We recognize that both 8 and 6 can be divided by their GCD, which is 2:

$$ \frac{8}{6} = \frac{8 \div 2}{6 \div 2} = \frac{4}{3} $$
A simplified fraction is always more straightforward to understand and work with. In our exercise, the simplified answer is \( x = \frac{4}{3} \).

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Most popular questions from this chapter

In the following exercises, solve the equation. $$ f+\left(-\frac{2}{3}\right)=\frac{5}{12} $$

In the following exercises, perform the indicated operation. $$ 8 \frac{6}{11}-2 \frac{9}{11} $$

In the following exercises, solve the equation. $$ y+\frac{3}{5}=\frac{7}{5} $$

In the following exercises, determine whether the each number is a solution of the given equation. $$ y+\frac{3}{5}=\frac{5}{9} $$ $$ (a)y=\frac{1}{2} $$ $$ (b)y=\frac{52}{45} $$ $$ (c)y=-\frac{2}{45} $$

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary. \(\frac{m+n}{p}\) when \(m=-6, n=-2, p=4\)
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