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Chapter 8: Problem 269

In the following exercises, solve the equation. $$ f+\frac{2}{3}=4 $$

Short Answer

Expert verified
The value of \( f \) is \( \frac{10}{3} \).

Step by step solution

01

Identify the equation

The given equation is $$ f + \frac{2}{3} = 4 $$
02

Isolate the variable

To find the value of the variable \( f \), isolate \( f \) on one side of the equation. To do this, subtract \( \frac{2}{3} \) from both sides of the equation:\[ f + \frac{2}{3} - \frac{2}{3} = 4 - \frac{2}{3} \]This simplifies to:\[ f = 4 - \frac{2}{3} \]
03

Simplify the right-hand side

Convert the whole number 4 to a fraction with the same denominator as \( \frac{2}{3} \):\[ 4 = \frac{12}{3} \]Now, subtract \( \frac{2}{3} \) from \( \frac{12}{3} \):\[ f = \frac{12}{3} - \frac{2}{3} \]
04

Perform the subtraction

Subtract the fractions on the right-hand side:\[ f = \frac{12 - 2}{3} \]This simplifies to:\[ f = \frac{10}{3} \]
05

Final answer

The value of \( f \) is:\[ f = \frac{10}{3} \]

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Key Concepts

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

Prealgebra
Prealgebra is the foundation of your mathematical journey. It covers the basics that you need for solving equations, which is what we're doing in this exercise. The goal is to get comfortable with numbers, variables, and simple operations.
This exercise, for example, asks us to solve an equation involving a variable and a fraction.
To solve for a variable means to determine what value it can take to make the equation true. Think of it as a puzzle where each piece (number or variable) must fit perfectly.
Prealgebra often serves as your introduction to concepts like fractions, variables, and basic operations, paving the way for more advanced topics.
Variable Isolation
To solve the equation, we need to isolate the variable on one side of the equation. In our case, we have the variable 'f'. Isolating a variable means getting it alone on one side of the equation.
To isolate 'f':
  • We start with the equation: \( f + \frac{2}{3} = 4 \).
  • Subtracting \( \frac{2}{3} \) from both sides helps move the constants to the other side: $$ f + \frac{2}{3} - \frac{2}{3} = 4 - \frac{2}{3} $$.
  • This simplifies to: \( f = 4 - \frac{2}{3} \).
By isolating the variable, we've set up the equation to easily solve for 'f'.
Fraction Subtraction
Subtracting fractions can be tricky but it becomes straightforward with practice. In this exercise, we need to subtract \( \frac{2}{3} \) from 4.
Steps to subtract fractions:
  • First, convert the whole number 4 to a fraction with the same denominator. Here, 4 becomes \( \frac{12}{3} \).
  • Subtract \( \frac{2}{3} \) from \( \frac{12}{3} \): \[ f = \frac{12}{3} - \frac{2}{3} \]
  • The numerators are then subtracted: \( f = \frac{12 - 2}{3} \).
  • This simplifies to: \( f = \frac{10}{3} \).
Fraction subtraction often involves finding a common denominator and then subtracting the numerators. In this case, we simplified the problem by converting the whole number to a fraction first.

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