Problem 27 - Prealgebra

Question

Simplify expression. \(8 x+5+7+2 x\)

Step-by-Step Solution

Answer

The simplified expression is \(10x + 12\).

Step 1: Identify Like Terms

First, identify the like terms in the expression. The expression we have is \(8x + 5 + 7 + 2x\). Like terms are terms that have the same variable raised to the same power. In this case, \(8x\) and \(2x\) are like terms because they both contain the variable \(x\). Similarly, \(5\) and \(7\) are like terms because they are constants.

Step 2: Combine Like Terms with Variables

Next, combine the like terms that contain the variable \(x\). Add \(8x\) and \(2x\) to get \(10x\). This simplifies the expression to \(10x + 5 + 7\).

Step 3: Combine Constant Terms

Now, combine the constant terms \(5\) and \(7\). Add these numbers to get \(12\). This further simplifies the expression to \(10x + 12\).

Step 4: Write the Final Simplified Expression

After combining all like terms, the simplified expression is \(10x + 12\). No further simplification can be done, as there are no more like terms to combine.

Key Concepts

What are Like Terms?

In algebra, identifying like terms is crucial for simplifying expressions. Like terms are terms in an expression that have the same variables raised to the same powers. They essentially "look alike" when it comes to the variables. Understanding this is the foundational step to simplifying any algebraic expression effectively.

Here's how you can spot like terms in any expression:

• Terms must contain the same variable(s). For instance, terms with the variable \(x\) will only be like terms if they all include \(x\).
• Variables must be raised to the same powers. This means \(x^2\) and \(x\) are not like terms because their exponents are different.
• Constants, numbers without variables, are also like terms. Simply, all the constant numbers in an expression categorize as like terms with each other.

Remember, the coefficient (the number in front of the variables) does not impact whether terms are like terms. It's solely about the variable parts matching up.

Combining Like Terms: Bringing Order to Chaos

Once you've identified your like terms, the next step is to combine them. Combining like terms is a simplified way of grouping and reducing them to clean up the expression. It's like tidying up a mathematical desk; similar items get grouped together, making them easier to manage and understand.

To combine like terms, follow these directions:

• Add or subtract the coefficients (the numbers in front of the variables) of like terms. Ensure you pay attention to any positive or negative signs.
• Keep the variable and its power the same. Forgetting this could lead to incorrect simplifications.
• Combining constant terms is a straightforward addition or subtraction.

For example, in the expression \(8x + 5 + 7 + 2x\), after identifying like terms, you can add \(8x\) and \(2x\) to get \(10x\). Then, combining the constants \(5\) and \(7\) gives \(12\). Thus, the expression becomes neat and simpler as \(10x + 12\).

The Art of Algebraic Expression Simplification

Simplifying algebraic expressions is critical for solving equations efficiently. The process of simplifying involves reducing an expression to its simplest form. This makes calculations easier and reduces the room for errors when working with complex equations.
To simplify an algebraic expression like \(8x + 5 + 7 + 2x\), use these tactics:

• First, identify and underline the like terms.
• Next, perform the operations – add or subtract – to combine the like terms.
• Ensure that you have accounted for every term in the expression. Double-checking your work can help avoid missing any details.

Remember, simplifying might seem repetitive at first, especially with straightforward expressions, but with complex equations, these steps can save you time and help prevent mistakes. Practice is key here; as you practice more, recognizing and combining like terms will become second nature, paving the way to more sophisticated problem solving.