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Chapter 2: Problem 24

Solve each formula for the indicated letter. $$ A=\frac{a+b+c}{3}, \text { for } b $$

Short Answer

Expert verified
b = 3A - a - c

Step by step solution

01

Original Formula

Start with the given formula:\[ A = \frac{a + b + c}{3} \]
02

Eliminate the Denominator

Multiply both sides of the equation by 3 to eliminate the denominator:\[ 3A = a + b + c \]
03

Isolate 'b'

Subtract 'a' and 'c' from both sides to solve for 'b':\[ b = 3A - a - c \]

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Key Concepts

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

Algebraic Manipulation
Algebraic manipulation involves performing operations on both sides of an equation to keep it balanced while simplifying or isolating a variable. This is important because it helps in finding the values of unknown variables. To manipulate equations correctly, keep these strategies in mind:
  • Perform the same operation on both sides of the equation
  • Use the inverse of any operations when moving terms from one side to the other
  • Combine like terms where possible to simplify expressions
In the given exercise, we first multiply both sides by 3 to eliminate the denominator, ensuring the equation stays balanced. Practicing these techniques will make understanding and solving algebraic equations more intuitive.
Formula Rearrangement
Rearranging a formula means rewriting it in a different way to isolate a specific variable. This often involves several algebraic manipulations. The skill of rearranging formulas is vital in solving equations across various math and science disciplines.
Here鈥檚 a general approach to rearrange formulas:
  • Identify the variable you need to isolate
  • Use algebraic manipulation to move other terms to the opposite side
  • Keep track of every operation to maintain equation balance
In our problem, we rearrange \( A = \frac{a + b + c}{3} \) to solve for \( b \). We first eliminate the fraction by multiplying by 3, resulting in \( 3A = a + b + c \. \) Then, by subtracting both \( a \) and \( c \) from both sides, we isolate \( b \) and reach \( b = 3A - a - c \).
Isolation of Variables
Isolating a variable means getting that variable alone on one side of the equation. This process makes it clear what the variable equals. It鈥檚 a crucial step in solving equations and inequalities. Follow this straightforward strategy:
  • Get rid of any fractions by multiplying
  • Use inverse operations to move terms
  • Simplify where possible
In the provided solution, we isolate \( b \) by strategically eliminating other terms. After multiplying by 3 and simplifying the equation \( 3A = a + b + c \), we subtract \('a' and 'c' from both sides \). This leaves \( b \) alone on one side of the equation: \ b = 3A - a - c \. By mastering variable isolation, students can make solving complex equations more manageable.

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