Problem 47 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ x-(-2)=6 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 4\).

Step 1: Simplify the Equation

The first thing to do is to simplify the equation by resolving the double negatives. This gives us \(x + 2 = 6\).

Step 2: Isolate the Variable

The next step is to isolate the variable 'x' on one side of the equation. We can do this by subtracting 2 from both sides of the equation. This leaves us with \(x = 6 - 2\).

Step 3: Find the Solution

By performing the operation on the right side of the equation, we find that \(x = 4\).

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that declare the equality of two expressions. They are comprised of variables, coefficients, constants, and an equals sign. For example, in the equation \(x - (-2) = 6\), \(x\) is the variable, \(6\) is the constant, and we don't explicitly see a coefficient, but it's understood to be 1 in front of \(x\). Equations are essential tools for modeling real-world problems and finding unknown values.

In this specific case, we're working with a linear equation, which is the simplest form of algebraic equations. Linear equations have variables raised to the power of one and graph as a straight line. The structure of these equations allows for straightforward methods to solve them, like the step-by-step solution we have here where our goal is to find the value of \(x\) that makes the equation true.

Isolating the Variable

Isolating the variable is a fundamental technique in algebra that involves manipulating an equation in such a way that the variable appears by itself on one side of the equals sign. To achieve this, you perform balanced operations on both sides of the equation to 'move' terms around without changing the equation's essence.

In our example, \(x - (-2) = 6\), the first step is to eliminate the double negative, resulting in \(x + 2 = 6\). Then, to isolate \(x\), we subtract 2 from both sides, giving us \(x = 6 - 2\), which simplifies to \(x = 4\). The order of operations is crucial for this process to be correctly executed. It ensures that we do not disrupt the balance of the equation and that each step follows logically from the last.

Resolving Double Negatives

Encountering double negatives in equations can be a little confusing at first, but they're pretty simple to resolve with basic arithmetic. A double negative occurs when we have two negative signs back-to-back, essentially meaning a negative of a negative value.

In the context of our exercise \(x - (-2) = 6\), we see the double negative with \(\-(-2)\). According to mathematical rules, two negatives multiply to produce a positive. Hence \(\-(-2)\) becomes \(+2\). This transformation simplifies our equation to \(x + 2 = 6\), making it more straightforward to solve. It's important to recognize double negatives as they significantly simplify the process of solving equations and understanding the nature of the relationships between terms within an equation.