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Chapter 2: Problem 119

Solve each equation. Be sure to check each answer. $$b+7=\frac{20}{3}$$

Short Answer

Expert verified
The solution to the equation is \(b=-\frac{1}{3}\)

Step by step solution

01

Isolate b

To isolate \(b\), subtract 7 from each side of the equation. \[b+7-7=\frac{20}{3}-7\] which simplifies to: \[b=\frac{20}{3}-\frac{21}{3}\]
02

Simplify the expression

Subtract the fractions to get the solution for \(b\): \[b=-\frac{1}{3}\]
03

Checking the solution

Substitute \(b=-\frac{1}{3}\) into the original equation to check if the left side equals the right side: \[(-\frac{1}{3})+7=\frac{20}{3}\] simplifying the left side gives us: \[ \frac{20}{3}=\frac{20}{3}\]

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Key Concepts

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

Linear Equations
Linear equations are fundamental in algebra. They help us find unknown values by evaluating relationships between numbers.
Linear equations can be expressed in the form \(ax + b = c\), where \(x\) is what we are solving for.
One must isolate the variable on one side of the equation to find its value.
  • This involves performing basic arithmetic operations, such as addition, subtraction, multiplication, and division.
  • The goal is to maintain equality by carrying out the same operation on both sides of the equation.
These operations are reversible, meaning that whatever actions we take to simplify an equation can, in theory, be reversed without error.
Linear equations are fundamental building blocks in mathematics, laying down the groundwork for complex subjects like calculus and statistics.
Fraction Operations
Working with fractions in equations requires precision, especially when they need to be added or subtracted.
Fractions may look complex, but they follow the same basic rules as whole numbers.
When adding or subtracting fractions, it's crucial to have a common denominator.
  • The denominator is the bottom part of a fraction, and it determines how many parts the whole is divided into.
  • When fractions have different denominators, convert them to equivalent fractions with a common denominator to make calculations possible.
For example, in the exercise, \(7\) is re-expressed as \(\frac{21}{3}\) to match the denominator of \(\frac{20}{3}\).
This allows you to perform the subtraction: \(\frac{20}{3} - \frac{21}{3} = -\frac{1}{3}\).
Understanding fraction operations is essential as they appear frequently not only in algebra but in real-world scenarios like cooking and construction, where precise measurements are necessary.
Isolation of Variables
Isolating the variable is the process of manipulating an equation to get the variable of interest by itself on one side.
This is a technique frequently employed in algebra to solve equations efficiently.
  • The very first step is picking a side of the equation to keep the variable.
  • Usually, this means performing operations like adding, subtracting, multiplying, or dividing.
In the given problem, isolating \(b\) involves removing the constant \(7\) from the left side.
Subtracting \(7\) from both sides effectively puts all terms except \(b\) on one side: \(b + 7 - 7 = \frac{20}{3} - 7\).
This simplifies to \(b = \frac{20}{3} - \frac{21}{3}\).
By mastering the isolation process, you simplify equations in a logical manner and become adept at problem-solving skills critical for higher math education and various professional fields.

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Most popular questions from this chapter

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