Problem 48 - Algebra 1: Concepts and Skills

Question

Solve the equation by simplifying first. $$ 2-(-b)=-6 $$

Step-by-Step Solution

Answer

The solution to the equation is \(b = -8\).

Step 1: Simplify The Equation

The first step is to simplify the equation by focusing on the double negatives, -(-b). In mathematics, subtracting a negative value is the same as adding a positive value: \(2 - (-b)\) simplifies to \(2 + b\). Therefore, the equation becomes \(2 + b = -6\).

Step 2: Isolate The Variable b

The next step is to isolate \(b\). We do this by subtracting 2 from both sides of the equation. Thus, \(b = -6 - 2\).

Step 3: Solve For b

Finally, we solve for \(b\). \(b = -6 - 2\) simplifies to \(b = -8\). Therefore, the solution to the equation is \(b = -8\).

Key Concepts

Simplify Expressions

Simplifying expressions is a fundamental skill in algebra. It involves making an expression easier to understand or solve by reducing it to its simplest form. In the exercise, the expression starts with a double negative: \(2 - (-b)\). Handling double negatives is crucial:

• Subtracting a negative is the same as adding a positive.
• So, \(2 - (-b)\) becomes \(2 + b\).

Simplifying helps clarify the equation, allowing us to see clearly what needs to be done next. Once simplified, the originally complicated-looking equation now looks neater and easier to manage: \(2 + b = -6\). Always aim to deal with any unnecessary complexity as a first step in solving equations.

Isolate Variables

Isolating variables is the process of rearranging an equation so that one variable stands alone on one side of the equals sign. This step is crucial when solving for the unknown. In the present exercise, after simplifying to \(2 + b = -6\), we isolate \(b\):

• Subtract 2 from both sides to counter the addition and move \(b\) to one side.
• This operation transforms the equation into \(b = -6 - 2\).

The goal here is to perform operations that keep the equation balanced while effectively centering attention on the variable we're solving for. Isolating variables helps create a clear path to the solution, making it evident what calculations remain.

Solve Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. After isolating \(b\), we arrived at the simplified linear equation \(b = -6 - 2\). To solve it:

• Perform the subtraction: \(-6 - 2\).
• The result is \(b = -8\).

Linear equations like this form the backbone of much mathematics, offering clear steps to arrive at a solution. The process provides a systematic approach to revealing the value of the unknown variable. By reducing expressions, isolating variables, and finally performing arithmetic operations, we ensure the solution process is both logical and efficient.