Problem 10 - Prealgebra

Question

Solve equation. Check your solution. \(2 x+3=x\)

Step-by-Step Solution

Answer

The solution is \(x = -3\).

Step 1: Move Variable Terms to One Side

We start with the equation: \[ 2x + 3 = x \]To get all variable terms on one side, subtract \(x\) from both sides: \[ 2x + 3 - x = x - x \]This simplifies to: \[ x + 3 = 0 \]

Step 2: Isolate the Variable

Now we need to solve for \(x\), so subtract 3 from both sides: \[ x + 3 - 3 = 0 - 3 \]This simplifies to: \[ x = -3 \]

Step 3: Check the Solution

To ensure our solution is correct, substitute \(x = -3\) back into the original equation:\[ 2(-3) + 3 = -3 \]Calculate the left side: \[ -6 + 3 = -3 \]The left side equals the right side, verifying that \(x = -3\) is indeed the correct solution.

Key Concepts

Understanding Algebraic Manipulation

Algebraic manipulation is a foundational skill in solving equations efficiently.
This involves rearranging and simplifying expressions to reveal the unknown variable. When working with linear equations, the goal is to isolate the variable on one side of the equation.
Let's break it down:

• Identify terms that contain the variable versus constant terms. This helps in recognizing what needs to be moved around.
• Add, subtract, multiply, or divide each term systematically. Ensure you do these operations on both sides of the equation to maintain equality.
• Simplify the equation step by step. Don't rush. Each operation should bring you closer to finding the variable's value.

In the original problem, we first subtracted \( x \) from both sides to align with our goal—simplifying the expression to get \( x + 3 = 0 \).
Understanding and practicing algebraic manipulation gives you the tools to solve equations methodically and confidently.

Importance of Equation Checking

Equation checking is like double-checking your homework. It's crucial to verify that your solution is correct by plugging it back into the original problem.
This simple process assures us that the solution makes sense and hasn't been derailed by a simple error. Here's how you can effectively check your solutions:

• Take your solution and substitute it back into the original equation.
• Perform the arithmetic operations to simplify both sides of the equation.
• Ensure both sides of the equation are equal after substitution. If they are, you've confirmed your solution.

In our provided example, we had \( x = -3 \). By substituting \( -3 \) back into the equation \( 2x + 3 = x \) and simplifying, both sides equaled \( -3 \). This affirmed that our solution was accurate.
It's always a good habit, helping avoid misunderstandings and increasing your reliability in math.

Techniques for Isolating Variables

Isolating variables is the heart of solving linear equations. It's about getting the unknown alone on one side of the equation.
Here's a step-by-step guide on how to isolate variables effectively:

• First, get all the terms with the variable on one side. You can add or subtract terms to do this. Ensure you move everything around equally on both sides of the equation.
• If other numbers are with the variable, divide or multiply to get the coefficient of the variable to one.
• Remove any constants by performing the opposite mathematical operation. For example, if there's a '+3' on the variable's side, you would subtract 3 from both sides.
• Continue these operations until the variable is by itself.

In our example, after the initial subtraction of \( x \), we had \( x + 3 = 0 \).
By subtracting 3 from both sides, we beautifully isolated \( x \), solving it as \( x = -3 \).
Mastering this concept allows for more complex problem-solving down the line, as it strengthens your equation solving base.