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Chapter 3: Problem 89

Solve. \(p t+p=w,\) for \(p\)

Short Answer

Expert verified
p = \frac{w}{t + 1}

Step by step solution

01

Identify the equation

The given equation is \(pt + p = w\). We need to solve for \(p\).
02

Factor out \(p\)

Notice that \(p\) is a common factor in the terms on the left-hand side. Factor \(p\) out: \[ p(t + 1) = w \]
03

Solve for \(p\)

To isolate \(p\), divide both sides of the equation by \(t + 1\): \[ p = \frac{w}{t + 1} \]

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Key Concepts

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

Factoring
Factoring is a critical skill in algebra that simplifies equations and makes them easier to solve. In the exercise, we started with the equation:
\( pt + p = w \).
Since both terms on the left-hand side share a common factor of \( p \), we can factor \( p \) out. This means rewriting the equation in a simpler form where \( p \) is outside of the parentheses:
\[ p(t + 1) = w \].
This step simplifies our expression and sets the stage for the next step in solving the equation.
When you factor, always look for common factors in each term. Factoring makes it much easier to isolate the variable you are solving for.
Isolating Variables
Isolating the variable is another vital part of solving equations. Here, we are solving for \( p \). After factoring out \( p \), our equation became:
\[ p(t + 1) = w \].
To isolate \( p \), we need to get \( p \) by itself on one side of the equation. This usually involves performing inverse operations to both sides of the equation.
In this case, we divide both sides of the equation by \( (t + 1) \):
\[ p = \frac{w}{t + 1} \].
Now, \( p \) is isolated, meaning it stands alone on one side of the equation. This method is a foundational skill that you will use repeatedly in algebra.
Algebraic Manipulation
Algebraic manipulation involves using various mathematical techniques to re-arrange equations and solve for variables. This includes operations like addition, subtraction, multiplication, division, and factoring.
In our problem, we used different forms of algebraic manipulation:
  • First, we identified the common factor of \( p \) and factored it out.
  • Then, we used division to isolate \( p \).

One important thing to remember is that whatever operation you do to one side of the equation, you must do to the other side. This keeps the equation balanced. For instance, if you divide by \( t + 1 \) on one side, you must also divide by \( t + 1 \) on the other side.
Mastering these skills will help you solve more complex equations in the future. Practice these steps until they become second nature.

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Most popular questions from this chapter

To prepare for Section \(3.5,\) review subtraction and order of operations. Simplify. $$ \frac{-3-5}{-1-(-1)} $$

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To prepare for Section \(3.8,\) review evaluating algebraic expressions (Section \(1.8)\) $$ n^{2}-5 n, \text { for } n=-1 $$
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