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Chapter 4: Problem 542

Mixed Practice, In the following exercises, solve. $$p+\frac{2}{3}=\frac{1}{12}$$

Short Answer

Expert verified
p = \(\frac{-7}{12}\)

Step by step solution

01

Isolate the variable term

Subtract \(\frac{2}{3}\) from both sides of the equation to isolate the term with the variable on one side. The equation becomes: \[ p + \frac{2}{3} - \frac{2}{3} = \frac{1}{12} - \frac{2}{3} \] Simplifying, we get: \[ p = \frac{1}{12} - \frac{2}{3} \]
02

Find a common denominator

To subtract the fractions, find a common denominator. The denominators are 12 and 3, so the common denominator is 12. Rewrite \(\frac{2}{3}\) as \(\frac{8}{12}\). The equation becomes: \[ p = \frac{1}{12} - \frac{8}{12} \]
03

Perform the subtraction

Subtract the fractions: \[ p = \frac{1}{12} - \frac{8}{12} = \frac{1 - 8}{12} = \frac{-7}{12} \]

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Key Concepts

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

Isolate the Variable
To solve a linear equation, one of the most important steps is to isolate the variable. In this equation, our goal is to get the variable, which is \( p \), by itself on one side of the equation.
We start with the equation:
$$p+\frac{2}{3}=\frac{1}{12}$$
We need to remove the fraction \( \frac{2}{3} \) from the left side. To do this, we subtract \( \frac{2}{3} \) from both sides of the equation. This way, we keep the equation balanced. Subtracting \( \frac{2}{3} \) from both sides looks like this:
$$p+\frac{2}{3} - \frac{2}{3} = \frac{1}{12} - \frac{2}{3}$$
This simplifies to: $$p = \frac{1}{12} - \frac{2}{3}$$
Now, we have isolated the variable on one side.
Common Denominator
When working with fractions, it's crucial to find a common denominator so you can easily add or subtract them. Let's focus on the fractions on the right side of the equation:
$$\frac{1}{12} - \frac{2}{3}$$
We need to rewrite \( \frac{2}{3} \) with a denominator of 12. The current denominators are 12 and 3. Since the least common multiple of 12 and 3 is 12, we can rewrite \( \frac{2}{3} \) as \( \frac{8}{12} \) (because \( 2 \times 4 = 8 \) and \( 3 \times 4 = 12 \)). Now, the equation becomes: $$p = \frac{1}{12} - \frac{8}{12}$$
Using a common denominator helps make the subtraction straightforward.
Fraction Subtraction
Once the fractions have a common denominator, subtracting them is simple. We start with:
$$p = \frac{1}{12} - \frac{8}{12}$$
To subtract these fractions, we subtract the numerators and keep the common denominator. So we get:
$$p = \frac{1-8}{12}$$
Simplifying the numerator gives:
$$p = \frac{-7}{12}$$
Now, we have our answer:
$$p = \frac{-7}{12}$$
By isolating the variable, finding a common denominator, and performing the fraction subtraction, we solved the equation and found the value of \( p \).

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Most popular questions from this chapter

In the following exercises, evaluate. $$ y-\frac{4}{5} $$ $$ (a)y=-\frac{4}{5} $$ $$ (b)y=\frac{1}{4} $$

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In the following exercises, add. $$\frac{2}{5}+\frac{1}{5}$$

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