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Chapter 2: Problem 144

Translate to an equation and then solve. -184 is the product of 23 and \(p\).

Short Answer

Expert verified
The value of \(p\) is -8.

Step by step solution

01

- Understand the Problem

The given statement says that -184 is the product of 23 and another number denoted by the variable \(p\). This means the equation will involve multiplication.
02

- Translate the Problem into an Equation

Translate the statement into a mathematical equation. 'The product of 23 and \(p\)' can be written as \(23 \cdot p\). Therefore, the equation is \(-184 = 23 \cdot p\).
03

- Solve the Equation for \(p\)

To find the value of \(p\), isolate \(p\). Divide both sides of the equation by 23: \(-184 / 23 = p\).
04

- Calculate the Value

Perform the division to get: \(-184 / 23 = -8\). So, \(p = -8\).

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Key Concepts

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

multiplication in equations
Multiplication appears often in equations, linking variables and constants. When you see 'the product of' in a problem, this means you'll multiply the given numbers. For example, in our exercise, 'the product of 23 and \( p \)' translates to \(23 \cdot p\). Here, 23 is multiplied by a variable \( p \), creating a product. Multiplication can help model real-world relationships in an equation format.

Understanding this helps you interpret phrases correctly. Remember, the order in multiplication doesn't matter: \(23 \cdot p\) is the same as \(p \cdot 23\). Practice translating similar phrases to sharpen your skills.
isolating variables
Isolating variables means getting the unknown variable on one side of the equation by itself. This is crucial for solving equations. Let's illustrate with our exercise.

We started with the equation \(-184 = 23 \cdot p\). To isolate \( p \), we need to remove the 23 from beside it. In this case, we do the opposite of multiplication, which is division.

Think of isolating the variable like undoing an equation step-by-step: if multiplication is applied, use division, and vice versa. This keeps the equation balanced, like a scale. Understanding how to isolate variables provides the foundation for solving more complex equations in algebra.
division in equations
When isolating a variable, division often comes into play. Consider our equation \(-184 = 23 \cdot p\). To solve for \( p \), we must get \( p \) alone. This means dividing both sides by 23. Thus:

\[ p = \frac{-184}{23} \]

Division helps simplify an equation to solve for the variable. It's important to perform division carefully, ensuring accuracy. Here, dividing -184 by 23 gives us \(-8\), so \( p = -8 \).

Division in equations reverses multiplication, vital in algebraic manipulation. Once you master this, you'll handle more equations confidently.

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