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Chapter 2: Problem 82

Solve each formula for the specified variable. $$ P=a+b+c \text { for } a $$

Short Answer

Expert verified
The solution is \(a = P - b - c\).

Step by step solution

01

Identify the given formula

The given formula is: \[ P = a + b + c \] We need to solve for the variable \(a\).
02

Isolate variable \(a\)

To isolate \(a\), subtract \(b\) and \(c\) from both sides of the equation: \[ P - b - c = a + b + c - b - c \]
03

Simplify the equation

Simplify the right-hand side of the equation by canceling out the terms \(b\) and \(c\), leaving: \[ a = P - b - c \]

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Key Concepts

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

Isolating Variables
Isolating a variable means you get that variable alone on one side of the equation. This process lets you find the value of the variable easily. To isolate a variable:
  • Identify the variable you need to solve for.
  • Move other terms to the opposite side of the equation.
It's like peeling off layers to get to the core. For instance, in this exercise, we need to solve for \(a\) in the formula \[P = a + b + c\]. By subtracting \(b\) and \(c\) from both sides, you isolate \(a\): \[a = P - b - c\]. Breaking it down step by step ensures you handle each term carefully without missing any detail.
Algebraic Manipulation
Algebraic manipulation involves using operations like addition, subtraction, multiplication, and division to change the form of an equation. Here's a breakdown of useful techniques:
  • Adding/Subtracting Terms: Move terms from one side to the other to isolate the variable.
  • Multiplying/Dividing: If the variable is multiplied/divided by a number, do the inverse operation to isolate it.
  • Simplification: Combine like terms and simplify the equation.
In our example:
1. Start with \[P = a + b + c\].
2. Subtract \(b\) and \(c\) from both sides: \[P - b - c = a + b + c - b - c\].
3. Simplify by eliminating \(b\) and \(c\) from the right: \[P - b - c = a\].
It's essential to follow logical steps and apply correct algebraic rules to maintain the equation's balance.
Formula Transformation
Formula transformation involves changing the original formula to solve for a different variable. This concept is widely used in both math and science. To transform a formula:
  • Select the target variable to solve for.
  • Use algebraic manipulation to isolate this variable.
Our initial formula is \[P = a + b + c\].
We want to transform it to solve for \(a\).
Following the steps to isolate \(a\):
1. Subtract \(b\) and \(c\) from both sides: \[P - b - c = a + b + c - b - c\].
2. Simplify to \[a = P - b - c\].
Now you have a new formula, \[a = P - b - c\], where \(a\) is expressed in terms of \(P\), \(b\), and \(c\). Understanding formula transformation is like learning to see equations from different angles, making it easier to solve various problems.

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