Problem 31 - Elementary Algebra

Question

Solve. $$x-1+3=0$$

Step-by-Step Solution

Answer

x = -2

Step 1: Equation Simplification

First, take the equation given: \(x-1+3=0\). Start by combining like terms. In this expression, \(-1 + 3 = 2\). Thus, the expression simplifies to \(x + 2 = 0\).

Step 2: Solve for x

Now that the equation is \(x + 2 = 0\), solve for \(x\). Subtract 2 from both sides to isolate \(x\). This gives \(x = -2\).

Key Concepts

Equation Simplification

Equation simplification is about making an equation easier to work with by combining like terms and reducing unnecessary complexity. In algebra, this often means gathering all constants together and simplifying them, leaving the variable terms clearer. Let's look at the original equation:

• Start with: \(x - 1 + 3 = 0\)
• Combine like terms: here \(-1\) and \(+3\) are both constants, so you add them
• Simplified: \(x + 2 = 0\)

This process not only cleans up the equation but sets the stage for solving it. Remember, the goal is to make the equation as straightforward as possible before trying to find the solution.

Solving Linear Equations

Solving linear equations requires finding the value of the variable where the equation holds true. A linear equation is one where the variable is raised only to the first power, resulting in a straight line graphically. This makes them predictable and relatively simple:

• Use simplification to make the equation clearer (as we did with \(x + 2 = 0\))
• Apply operations to isolate the variable on one side of the equation

This step is essential in ensuring all linear equations are solved seamlessly. By isolating terms and constants methodically, achieving the solution becomes a focused task. Linear equations typically require just a few steps to arrive at the answer, demonstrating their straightforward nature.

Isolate Variable

Isolating the variable is a critical step in solving equations. This involves moving all terms with the variable to one side of the equation and the constants to the other. The objective is often to have the variable by itself:

• In \(x + 2 = 0\), we aim to get \(x\) alone
• To achieve this, subtract \(2\) from both sides (whatever you do to one side of the equation, you must do to the other)
• The equation, once simplified, results in \(x = -2\)

Isolating the variable makes the equation transparent, revealing the value it takes to satisfy the equation. Learning to isolate efficiently is foundational, setting the basis for more complex algebraic manipulations in the future.