Problem 33 - Algebra A Combined Function

Question

Solve each equation. See Examples 9 and \(10 .\) \(8 x+20=6 x+18\)

Step-by-Step Solution

Answer

The solution is \(x = -1\).

Step 1: Collect Like Terms

First, we need to get all the terms involving \(x\) on one side of the equation and the constant numbers on the other side. Let's start by subtracting \(6x\) from each side to eliminate \(x\) on the right side: \ 8x + 20 - 6x = 6x + 18 - 6x. \ This simplifies to: \ 2x + 20 = 18.

Step 2: Isolate the Variable

Next, we need to isolate \(x\) by getting rid of the constant term on the left side of the equation. To do this, subtract \(20\) from both sides: \ 2x + 20 - 20 = 18 - 20. \ This results in: \ 2x = -2.

Step 3: Solve for x

In this step, we solve for \(x\) by dividing both sides by \(2\) to get \(x\) alone: \ \frac{2x}{2} = \frac{-2}{2}. \ Simplifying this gives: \ x = -1.

Key Concepts

Collecting Like Terms

When solving linear equations, one of the first steps is collecting like terms. This process involves grouping similar terms together on one side of the equation. Similar terms are those that have the same variable raised to the same power. For example, in the equation \(8x + 20 = 6x + 18\), both \(8x\) and \(6x\) are like terms because they both involve the variable \(x\).

Here's how to collect like terms effectively:

• Identify all terms that include the variable \(x\). Place them on one side of the equation. In our example, we subtract \(6x\) from both sides to do this.
• Combine the terms by performing the subtraction: \(8x - 6x\) becomes \(2x\).
• Ensure constants, such as \(20\) on the left and \(18\) on the right, are grouped together separately.

Understanding how to collect like terms simplifies your equations and makes them easier to solve.

Isolating the Variable

After collecting like terms, the next step is to isolate the variable you are solving for—in this case, \(x\). Isolating the variable means getting it by itself on one side of the equation. This is crucial for finding the solution because it clarifies what the variable equals.

For instance, consider the equation \(2x + 20 = 18\) after we've collected like terms:

• Target the constant number on the side of the equation that contains \(x\). Here, the number \(20\) is next to \(2x\).
• Subtract this constant from both sides to remove it: \(20 - 20\) leaves \(2x = 18 - 20\).
• After performing the subtraction, simplify to obtain \(2x = -2\).

By isolating \(x\), you're preparing to solve the equation. This step is essential for making complex equations more manageable.

Mathematical Operations

The final part of solving the equation involves using mathematical operations to find the value of the isolated variable. These operations are essential tools in manipulating the equation to reveal the solution.

From our example, we need to solve \(2x = -2\):

• To isolate \(x\) completely, divide each side of the equation by \(2\). This operation reverses the multiplication of \(x\) by \(2\).
• Execute the division: \(\frac{2x}{2} = \frac{-2}{2}\).
• This simplifies to \(x = -1\).

Key mathematical operations, like addition, subtraction, multiplication, or division, allow you to transform and work through equations systematically. Each operation has its role in addressing different parts of an equation, ultimately helping you find the correct solution with precision.