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Chapter 1: Problem 71

Solve for the specified variable. (See Examples \(8-9)\) $$ P=a+b+c \text { for } c $$

Short Answer

Expert verified
c = P - a - b

Step by step solution

01

Understand the problem

The equation provided is \(P = a + b + c\) and the goal is to solve for the variable \(c\). This means that we need to express \(c\) in terms of the other variables and constants.
02

Isolate the variable

Rearrange the equation \(P = a + b + c\) to isolate \(c\). Begin by subtracting \(a\) and \(b\) from both sides of the equation:\[P - a - b = a + b + c - a - b\]This simplifies to:\[P - a - b = c\]
03

Solution

The variable \(c\) is now isolated. Thus, the solution for \(c\) in terms of \(P\), \(a\), and \(b\) is:\[c = P - a - b\]

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Key Concepts

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

algebraic equations
An algebraic equation is a mathematical statement that shows the relationship between different variables and constants. In this type of equation, we use algebra to find unknown values. Algebraic equations can be simple, like our example, or very complex. When solving an algebraic equation, we often need to perform several operations such as addition, subtraction, multiplication, or division to find the value of an unknown variable. In our example, the given equation is

\( P = a + b + c \).

Here, the goal is to rearrange this equation to solve for the variable c.
isolate variable
To solve for a particular variable, we need to 'isolate' it. This means we want to get that variable alone on one side of the equation. By doing so, we can clearly see its value in terms of the other variables and constants in the equation. Let's see how we isolate c in the equation

\( P = a + b + c \).

First, identify the variable you need to isolate, which is c. Next, we need to 'move' the other terms to the other side of the equation. Since a and b are added to c, we'll subtract them from both sides:

\( P - a - b = c \).

Now c is isolated on the right side, and we have successfully solved for c.
algebraic manipulation
Algebraic manipulation involves using algebraic operations to simplify or rearrange equations. It is a fundamental skill in algebra and is essential for solving equations. In our example, we used subtraction to isolate the variable c.

Let's summarize the steps we took:

  • We started with the equation
    \( P = a + b + c \).

  • We needed to solve for c, so we decided to isolate it.
  • We subtracted a and b from both sides of the equation to get
    \( P - a - b = c \).

This subtraction is an example of algebraic manipulation. Such techniques are used in various algebraic problems to simplify and find solutions.

It is important to practice these skills to become comfortable with manipulating equations to solve for any variable.

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Most popular questions from this chapter

Write an inequality that represents the statement. The distance between \(x\) and \(c\) is less than \(\delta\) and greater than \(0(\delta\) is the lowercase Greek letter "delta").

a. Write an absolute value equation or inequality to represent each statement. b. Solve the equation or inequality. Write the solution set to the inequalities in interval notation. The distance between a number \(x\) and 3 on the number line is 8 .

a. Write an absolute value equation or inequality to represent each statement. b. Solve the equation or inequality. Write the solution set to the inequalities in interval notation. The variation between the measured value \(v\) and \(16 \mathrm{oz}\) is less than 0.01 oz.

Solve the equation or inequality. Write the solution set to each inequality in interval notation. a. \(|b+1|-4=1\) b. \(|b+1|-4 \leq 1\) c. \(|b+1|-4 \geq 1\)

Solve the equations. \(\left|\frac{1}{4} w\right|=|4 w|\)
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