Problem 22 - Algebra 1

Question

Solve the equation algebraically. Check your solution graphically. $$-x=-2$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 2\).

Step 1: Rewrite the Equation

Rewrite the original equation without the negative signs. This would make it easier to solve. So, \[-x = -2 \]can be written as \[x = 2 \]

Step 2: Solve the Equation

The term x has been isolated on one side, meaning that 2 is our solution. Using a number line, 2 is marked to graphically represent the solution.

Key Concepts

Algebraic Manipulation

When it comes to solving algebraic equations, understanding algebraic manipulation is essential. This ability enables you to transform complex expressions into simpler ones, eventually isolating the variable of interest. Think of algebraic manipulation as a toolbox containing different tools like the properties of equality, distributive property, and combining like terms.

Take the equation \( -x = -2 \) from our original exercise. The goal is to simplify the equation to make the solution evident. Here, manipulation involves recognizing that multiplying or dividing both sides of an equation by -1 will not change its solution. Thus, by applying this tool, we change \( -x = -2 \) into \( x = 2 \), a much simpler equation with a clear solution.

Graphical Solution Representation

A graphical solution representation offers a visual approach to understanding an algebraic solution. By representing equations on a graph or number line, you can see the practical implications of your algebraic work. This method is particularly helpful for visual learners and makes abstract algebraic concepts concrete.

After isolating \( x \) in the equation \( x = 2 \), the solution is intuitively checked by plotting it on a number line. This graphically confirms that \( x = 2 \) is indeed the point where the function \( y = -x \) would intersect the horizontal axis, affirming that the algebraic manipulation led to a correct and verifiable solution.

Isolating Variables

Isolating variables is a fundamental skill in algebra that involves manipulating the equation to get the variable you're solving for by itself on one side of the equation. This process makes it much easier to see the solution. In our exercise, the original equation \( -x = -2 \) already presents a situation where isolating the variable is straightforward.

The variable \( x \) is 'isolated' by removing the negative sign from both sides. With this equation, no operations such as addition, subtraction, multiplication, or division are necessary beyond this point. Thus, isolating variables can often be one of the simplest yet most important steps in the process of solving algebraic equations.