Problem 3 - Algebra A Combined Function

Question

Solve each equation. Check each solution. $$ x-2=-4 $$

Step-by-Step Solution

Answer

The solution is \( x = -2 \).

Step 1: Isolate the variable

We start by solving the equation \( x - 2 = -4 \) by isolating \( x \). To do this, we need to add 2 to both sides of the equation. This gives us: \[ x - 2 + 2 = -4 + 2 \] Simplifying both sides, we get: \[ x = -2 \]

Step 2: Check the solution

To ensure that \( x = -2 \) is indeed a solution to the original equation, we substitute \( -2 \) back into the equation \( x - 2 = -4 \). Substituting gives: \[ -2 - 2 = -4 \] Simplifying, we have: \[ -4 = -4 \] Since both sides are equal, \( x = -2 \) is the correct solution.

Key Concepts

Isolating Variables

When dealing with linear equations, one of the first steps is often to isolate the variable. The goal of isolating variables is to have your target variable on one side of the equation while everything else goes to the other side. This involves using basic arithmetic operations such as addition, subtraction, multiplication, and division.
In the given example, we start with the equation \( x - 2 = -4 \). To isolate \( x \), we need to perform operations that will leave \( x \) by itself on one side of the equation. Since the equation involves subtracting 2 from \( x \), we do the opposite operation to both sides of the equation — that is, we add 2. This process looks like:

• Adding 2 to both sides results in \( x - 2 + 2 = -4 + 2 \)
• This simplifies to \( x = -2 \)

Remember that whatever operation you do to one side, you must do to the other side to maintain equality. Isolating variables is a crucial skill in algebra as it lays the groundwork for solving more complex equations.

Checking Solutions

Once you have found a solution to an equation, it's important to verify that it is correct. This step ensures accuracy, and is especially useful in a learning environment. In our solved example, we found that \( x = -2 \). To check this solution, we substitute it back into the original equation to see if both sides of the equation remain equal.
The original equation was \( x - 2 = -4 \). By substituting \( -2 \) for \( x \), we perform the following steps:

• \( -2 - 2 \)
• This simplifies to \( -4 \)
• The equation now reads \( -4 = -4 \), which is true

Both sides of the equation equate, confirming that \( x = -2 \) is indeed the correct solution. Checking solutions can prevent errors and help build confidence in your answer.

Substitution Method

The substitution method is a powerful technique in algebra especially useful when dealing with systems of equations. Though you are only dealing with a single equation in this instance, the principle remains the same. You replace variables with known values to verify or find solutions.
In the context of our solved equation \( x - 2 = -4 \), after isolating \( x \) and finding it to be \( -2 \), we used substitution to check our answer. By replacing \( x \) with \( -2 \), the equation becomes a simple arithmetic problem:

• Substitute \( -2 \) in the equation: \( -2 - 2 = -4 \)
• Simplify: \( -4 = -4 \), confirming the equality

This method is essential in verifying solutions and is typically used in more complex problems to find unknowns when given other equations to work with. Understanding how to substitute correctly can greatly assist in solving not just linear equations, but also more advanced mathematical problems.