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Chapter 8: Problem 171

Solve the linear equation using the general strategy. $$8(3+3 p)=0$$

Short Answer

Expert verified
The solution is \( p = -1 \).

Step by step solution

01

- Distribute the Constant

Apply the distributive property to remove the parentheses. Multiply 8 by both terms inside the parentheses: \[ 8(3 + 3p) = 0 \] becomes \[ 8 \times 3 + 8 \times 3p = 0 \].
02

- Simplify the Equation

Simplify the equation from Step 1 by performing the multiplications: \[ 8 \times 3 = 24 \] and \[ 8 \times 3p = 24p \]. The equation is now: \[ 24 + 24p = 0 \].
03

- Isolate the Variable Term

To isolate the term with the variable, subtract 24 from both sides of the equation: \[ 24 + 24p - 24 = 0 - 24 \]. This simplifies to \[ 24p = -24 \].
04

- Solve for p

Divide both sides of the equation by 24 to solve for \( p \): \[ p = \frac{-24}{24} \]. This simplifies to \[ p = -1 \].

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Key Concepts

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

Distributive Property
In algebra, the distributive property is a powerful tool that allows you to multiply a single term across terms inside parentheses. When you see an expression like \(8(3 + 3p) = 0\), you can distribute the 8 to both 3 and 3p.
This means you multiply 8 by each term inside the parentheses: 8 times 3, and 8 times 3p.
This transforms the equation into: \[8 \times 3 + 8 \times 3p = 0\].
It's important to ensure you multiply correctly; in this case: \(8 \times 3 = 24\) and \(8 \times 3p = 24p\).
As a result, the equation becomes: \[24 + 24p = 0\].
Using the distributive property correctly simplifies your work and brings clarity to complex algebraic expressions.
Variable Isolation
Once you've simplified the equation, the goal is to isolate the variable to solve for it. In the equation \[24 + 24p = 0\], the term we want to isolate is 24p.
To do so, subtract 24 from both sides of the equation: \[24 + 24p - 24 = 0 - 24\].
This simplifies to \[24p = -24\].
Why does this work? You're balancing the equation. Removing 24 from the left side means you must do the same on the right to keep it equal.
Now you've successfully isolated 24p on one side, the next step is to divide by 24, which solves for p.
Simplifying Algebraic Expressions
After isolating the variable, you often need to simplify the resulting expression to find the value of the unknown. We left off at \[24p = -24\].
To solve for p, divide both sides by the coefficient of p, which is 24: \[p = \frac{-24}{24}\].
This simplifies to: \[p = -1\].
It's crucial to divide correctly; mistakes here can lead to wrong answers.
By simplifying algebraic expressions, we reduce equations to their simplest form, making it easier to understand and solve them.
Remember to check your solutions by substituting them back into the original equation. This ensures the left side equals the right side, verifying correctness.

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