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Chapter 15: Problem 30

Solve. $$ b+(b+5)-2(b-5)=18+b $$

Short Answer

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Step by step solution

01

- Distribute the negative sign

First, distribute the \(-2\) to both terms inside the parenthesis: \( b + (b + 5) - 2(b - 5) = 18 + b \) becomes \( b + b + 5 - 2b + 10 = 18 + b \)
02

- Combine like terms

Next, combine like terms on the left side: \( b + b - 2b + 5 + 10 = 18 + b \) simplifies to \(3\)
03

- Simplify the equation

Combine the constants on the left side: \(0 + 15 = 18 + b \) simplifies to \(0 = 3 + b \)
04

- Solve for b

Isolate \( b \) by moving it to the left side of the equation: \( 0 = 3 \)

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Key Concepts

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

Distributive Property
To solve the exercise, we first need to use the distributive property. This principle states that multiplying a number by a sum is the same as multiplying each addend by the number and then adding the results. In the exercise, we distribute the -2 through the parenthesis:
$$ -2(b - 5) = -2 * b + -2 * -5 $$
This simplifies to:
$$ -2b + 10 $$
Using the distributive property ensures that every term inside the parenthesis is correctly accounted for in the equation.
Combining Like Terms
Once the distributive property is applied, the next step is to combine like terms. Like terms are terms that contain the same variable raised to the same power. In the equation:
$$ b + (b + 5) - 2(b - 5) = 18 + b $$
After distributing:
$$ b + b + 5 - 2b + 10 = 18 + b $$
To combine like terms, group them together:
$$ (b + b - 2b) + (5 + 10) = 18 + b $$
Simplify it down to:
$$ 0 + 15 = 18 + b $$
Combining like terms simplifies the equation, making it easier to isolate variables and solve for them.
Isolating Variables
The final step in solving the equation is isolating the variable. This means to get the variable by itself on one side of the equation. Starting from:
$$ 0 + 15 = 18 + b $$
We simplify this to:
$$ 15 = 18 + b $$
To isolate b , subtract 18 from both sides:
$$ 15 - 18 = b $$
Simplify:
$$ -3 = b $$
Isolating the variable allows us to find the value of b. Always double-check your work to ensure accuracy.

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