minus 2 b minus left parenthesis minus 3 b minus 1 right parenthesis equals b minus 3 left parenthesis 2 minus b right parenthesis plus 1

Let's break down and solve the equation step by step:

The given equation is:

\[
-2b - \left( -3b - 1 \right) = b - 3\left(2 - b\right) + 1
\]

### Step 1: Simplify both sides of the equation

**Left-hand side:**

Distribute the negative sign across the parentheses:
\[
-2b - (-3b - 1) = -2b + 3b + 1
\]
Simplify:
\[
b + 1
\]

**Right-hand side:**

Distribute \(-3\) across the parentheses:
\[
b - 3(2 - b) + 1 = b - 6 + 3b + 1
\]
Simplify:
\[
4b - 5
\]

### Step 2: Set the simplified expressions equal to each other

Now, we have:
\[
b + 1 = 4b - 5
\]

### Step 3: Solve for \( b \)

Subtract \( b \) from both sides:
\[
1 = 3b - 5
\]

Add 5 to both sides:
\[
6 = 3b
\]

Divide both sides by 3:
\[
b = 2
\]

### Final Answer:
\[
b = 2
\]

Do you want further details or have any questions?

Here are some follow-up questions:
1. What if the equation contained fractions?
2. Can you solve an equation with two variables?
3. How do you solve quadratic equations?
4. What happens when there's no solution to an equation?
5. How can we check if the solution is correct?

**Tip:** Always check your solution by substituting the value of the variable back into the original equation to verify its correctness.

Learn how to solve the linear equation -2b - (-3b - 1) = b - 3(2 - b) + 1 with a detailed step-by-step solution. This problem covers key algebraic concepts like the distributive property, linear equations, and simplification. Suitable for students in Grades 8-10.

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