Problem 41 - Algebra A Combined Function

Question

Solve. $$ 3 x-6=2 x+5 $$

Step-by-Step Solution

Answer

The solution is \( x = 11 \).

Step 1: Set up the equation

You are given the equation: \[ 3x - 6 = 2x + 5 \]. We need to find the value of \( x \) that satisfies this equality.

Step 2: Subtract \( 2x \) from both sides

To get all \( x \)-terms on one side, subtract \( 2x \) from both sides of the equation: \[ 3x - 2x - 6 = 2x - 2x + 5 \]. Simplifying this gives \[ x - 6 = 5 \].

Step 3: Add 6 to both sides

Now, eliminate the constant term on the left side by adding 6 to both sides: \[ x - 6 + 6 = 5 + 6 \]. This simplifies to \[ x = 11 \].

Step 4: Verify the solution

Substitute \( x = 11 \) back into the original equation to check your work: \[ 3(11) - 6 = 2(11) + 5 \]. This simplifies to \[ 33 - 6 = 22 + 5 \] or \[ 27 = 27 \]. Since both sides of the equation are equal, \( x = 11 \) is indeed the correct solution.

Key Concepts

Step-by-step solutions

Solving linear equations can seem daunting, but breaking it down into simple steps makes the process much more manageable. First, we set up the equation properly. In our example, the equation is given as \( 3x - 6 = 2x + 5 \). To find the value of \( x \), we aim to isolate \( x \) on one side of the equation.

Here's how you can solve it in a structured manner:

• Step 1: Keep the original equation in mind: \( 3x - 6 = 2x + 5 \).
• Step 2: Move all terms containing \( x \) to one side. By subtracting \( 2x \) from both sides, we get \( x - 6 = 5 \).
• Step 3: Next, handle the constant term. Add 6 to both sides, resulting in \( x = 11 \).
• Step 4: Finally, verify the solution to ensure accuracy.

This methodical approach helps maintain accuracy and understanding, providing a clear path to solving each problem.

Equation simplification

Simplifying the equation is a crucial step in finding the solution. During simplification, we focus on reducing the equation to its simplest form so we can easily identify the value of unknown variables like \( x \).

In the given exercise, the first action was to subtract \( 2x \) from both sides. This is known as combining like terms, a fundamental part of simplification. So from \( 3x - 6 = 2x + 5 \), we moved to \( x - 6 = 5 \) by reducing the terms on both sides. This keeps our equation less cluttered and more straightforward.

Next, handling constant terms is important for reaching the solution. By adding 6 to both sides, \(-6\) on the left cancels out, giving us \( x = 11 \). The ability to simplify properly ensures we are on the right track to solving the equation efficiently.

Verifying solutions

Verifying solutions is a key step in problem-solving, especially with equations. It confirms that the solution we obtained is indeed correct, making sure we haven’t made any mistakes during simplification.

In our problem, once we have \( x = 11 \), we plug it back into the original equation to check our result: \( 3x - 6 = 2x + 5 \). Substituting \( x = 11 \), the equation becomes \( 3(11) - 6 = 2(11) + 5 \). Simplifying both sides gives \( 33 - 6 = 22 + 5 \) or \( 27 = 27 \).

Since both sides are equal, our solution \( x = 11 \) is validated. This step acts as a final check to ensure accuracy and bolsters confidence in your solving process, knowing that each calculation was executed correctly.