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Chapter 8: Problem 274

In the following exercises, solve the equation. $$ 8(3 p+5)-23(p-1)=35 $$

Short Answer

Expert verified
p = -28

Step by step solution

01

- Distribute the Constants

Distribute the constants 8 and -23 across the terms inside the parentheses:\[ 8(3p + 5) - 23(p - 1) = 35 \ 24p + 40 - 23p + 23 = 35 \]
02

- Combine Like Terms

Combine the like terms on the left-hand side of the equation:\[ 24p - 23p + 40 + 23 = 35 \ p + 63 = 35 \]
03

- Isolate the Variable

Subtract 63 from both sides of the equation to solve for p:\[ p + 63 - 63 = 35 - 63 \ p = -28 \]

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Key Concepts

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

Distributive Property
The distributive property is a powerful tool in simplifying expressions and solving equations. It allows you to multiply a single term by each term inside parentheses.
In our exercise, we applied the distributive property twice: once with 8 and once with -23.
Here's how:
\[ 8(3p + 5) - 23(p - 1) = 35 \]
When we distribute 8 across \(3p + 5\), we get:
\[ 8 \times 3p + 8 \times 5 = 24p + 40 \]
And when we distribute -23 across \(p - 1\), we get:
\[ -23 \times p - 23 \times (-1) = -23p + 23 \]
It's crucial to remember that the sign in front of the number being distributed affects all terms inside the parentheses.
Combining Like Terms
After distributing, you often need to combine like terms to simplify the equation. Like terms are terms that have the same variable raised to the same power.
For example, in our equation \( 24p + 40 - 23p + 23 = 35 \), both \(24p\) and \(-23p\) are like terms because they both contain the variable \(p\).
Similarly, \(40\) and \(23\) are like terms because they are constants.
Combining them gives:
\[ 24p - 23p + 40 + 23 = p + 63 \]
This simplification makes it much easier to isolate the variable and solve the equation.
Isolating Variables
The final step in solving a linear equation is isolating the variable. This means getting the variable (in our case, \(p\)) by itself on one side of the equation. We started with:
\[ p + 63 = 35 \] To isolate \(p\), we need to get rid of the \(+63\). We do this by performing the opposite operation, which is subtraction:
\[ p + 63 - 63 = 35 - 63 \]
Simplifying both sides, we get:
\[ p = -28 \] Now, \(p\) is isolated, and we have our solution. Always remember to perform the same operation on both sides of the equation to maintain the balance.

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