Question

Solve the equation. $$-b=8$$

Short answer

The solution is \(b = -8\).

Step-by-step solution

Interpreting the problem

We have the equation \(-b = 8\). We want to find out what \(b\) is, so we will need to isolate \(b\) on one of the sides of the equation.

Solving the equation

In order to get \(b\) alone on one side, we can multiply or divide both sides by \(-1\). This does not change the equation, it simply changes the sign on both sides. Therefore \(b = -8\).

Key concepts

Isolating Variables

When solving linear equations, one of the most fundamental skills you'll need is isolating variables. This process involves rearranging an equation so that the variable we're solving for, usually represented by a letter like 'x' or 'b', is on one side of the equation all by itself.

Consider the simple equation from our exercise, \( -b = 8 \). The goal is to get \( b \) isolated. This means we want \( b \) to be the only thing on its side of the equal sign, making it easy to read off the solution. Isolating variables becomes second nature with practice, and it's the cornerstone for solving more complex algebraic equations down the line.

Algebraic Manipulation

Algebraic manipulation is key to mastering algebra. It consists of various techniques used to transform equations into simpler forms or to isolate variables. These techniques include, but are not limited to, adding, subtracting, multiplying, and dividing both sides of an equation by the same number, which preserves the equation's balance.

In the exercise \( -b = 8 \), our variable \( b \) is currently multiplied by -1. To isolate \( b \) and to simplify the equation, we would perform an algebraic manipulation. By multiplying or dividing both sides by -1, we can flip the sign on both sides, leading to the simple and clear solution \( b = -8 \). This kind of algebraic thinking is essential in all areas of mathematics and can be applied to increasingly complex problems.

Inverse Operations

Understanding inverse operations is a foundational element in solving equations. An inverse operation reverses the effect of another operation. For example, addition and subtraction are inverses of each other, as are multiplication and division. By applying inverse operations, we can effectively 'undo' an operation to help isolate a variable within an equation.

In our example \( -b = 8 \), the inverse operation used to isolate \( b \) is multiplication by -1 because it is the inverse of multiplying by -1, which has already occurred. This action negates the initial multiplication by -1, leaving us with \( b = -8 \). Knowing when and how to use inverse operations correctly is a valuable skill that allows students to simplify and solve equations with confidence.