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Chapter 2: Problem 88

Solve each formula for the specified variable. \(P=a+b+c\) for \(a\)

Short Answer

Expert verified
a = P - b - c

Step by step solution

01

Understand the given formula

The given formula is: \( P = a + b + c \)It represents the sum of the variables \(a\), \(b\), and \(c\).
02

Rearrange the formula to solve for \(a\)

To solve for \(a\), isolate \(a\) on one side of the equation. Start by subtracting \(b\) and \(c\) from both sides of the equation:\[ P - b - c = a + b + c - b - c \]
03

Simplify the equation

Simplify the right side of the equation by canceling out \(b\) and \(c\). This leaves:\[ a = P - b - c \]
04

Write the final rearranged formula

The final equation solving for \(a\) is:\[ a = P - b - c \]

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Key Concepts

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

Algebraic Manipulation
Algebraic manipulation is a crucial concept in solving equations. It involves performing various operations to alter the form of an equation without changing its fundamental meaning.
These operations include addition, subtraction, multiplication, and division, which can help in simplifying and rearranging equations.
Consider the exercise given:
  • Given formula: \( P = a + b + c \)
  • We aim to solve for \( a \).
By using algebraic manipulation, you subtract \( b \) and \( c \) from both sides of the equation. This action keeps the equality intact but changes the form of the equation, allowing us to isolate the variable we are focusing on.
Rearranging Formulas
Rearranging formulas is a method used to express a specific variable in terms of other variables in the equation.
This process often involves multiple steps of algebraic manipulation to isolate the desired variable.
In our example, to solve for \( a \) in the formula \( P = a + b + c \), we use these steps:
  • Understanding that \( P \) equals the sum of \( a \), \( b \), and \( c \).
  • Identifying that our goal is to have \( a \) on one side.
  • Subtracting \( b \) and \( c \) from both sides results in: \[ P - b - c = a \]
By strategically rearranging the formula using these steps, we express \( a \) explicitly in terms of \( P \), \( b \), and \( c \).
Isolating Variables
Isolating variables involves rearranging the equation so that the variable you need to solve for is alone on one side of the equation.
This step-by-step process uses algebraic operations to achieve this goal.
For the given exercise:
  • We start with \( P = a + b + c \).
  • To isolate \( a \), subtract \( b \) and \( c \) from both sides: \[ P - b - c = a \]
  • This manipulation leaves \( a \) alone on one side, resulting in \[ a = P - b - c \].
The goal is to have the variable of interest (in this case \( a \)) standing by itself. By isolating \( a \), we can understand how it relates to the other terms in the equation.

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Most popular questions from this chapter

The numbers on two consecutive checkbook checks have a sum of 357 . What are the numbers?

U.S. five-cent coins are made from a combination of nickel and copper. For every \(1 \mathrm{lb}\) of nickel, 3 lb of copper are used. How many pounds of each metal would be needed to make 560 lb of five-cent coins? (Data from The United States Mint.)

To see the connection between the solution of an equation and the solutions of the corresponding inequalities. Solve the following inequality, and graph the solution set on a number line. \( 3 x+2>14 \)

The Toyota Camry and the Honda Civic were top-selling passenger cars in the United States in 2016 . Civic sales were 22 thousand less than Camry sales, and 756 thousand of the two cars were sold. How many of each car were sold? (Data from www.businessinsider.com)

To see the connection between the solution of an equation and the solutions of the corresponding inequalities. Solve the following inequality, and graph the solution set on a number line. \( 3 x+2<14 \)
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