Problem 4 - Prealgebra

Question

Simplify expression. \(6 a+4+2 a\)

Step-by-Step Solution

Answer

The simplified expression is 8a + 4.

Step 1: Combine Like Terms

In this expression, we have terms involving the variable 'a' and constant terms. First, we identify and combine like terms. The terms with 'a' are: 6a and 2a. We add these coefficients together: 6 + 2 = 8. So, the combined term is 8a.

Step 2: Simplify Constant Terms (if any)

In the given expression, there is only one constant term, which is the number 4. Since there are no other constant terms to combine it with, it remains the same.

Step 3: Write the Simplified Expression

After combining like terms, the expression simplifies to 8a + 4. Make sure all like terms are combined correctly and constants are appropriately added to the final expression.

Key Concepts

Combining Like Terms

When it comes to algebraic expressions, one of the most fundamental steps is combining like terms. But what exactly are like terms? They are terms in an expression that have the same variables raised to the same powers. Simply put, they share the same factor, allowing them to be combined. For instance, in the expression \(6a + 4 + 2a\), the terms \(6a\) and \(2a\) are like terms because both involve the variable 'a'. By adding these together, you simplify the expression, making it easier to work with.
Combining involves:

• Identifying terms with the same variable factor.
• Adding or subtracting their coefficients.

After combining, you reduce the number of terms, simplifying computations. So, in the exercise given, combining \(6a\) and \(2a\) results in \(8a\). Always look for like terms first when simplifying expressions.

Constants in Algebra

Constants are the numbers in algebraic expressions that don't change. They are called constants because, unlike variables, they keep their specific value no matter what. Understanding how to work with constants is crucial for simplifying expressions efficiently.
In the expression \(6a + 4 + 2a\), the number 4 is a constant. Unlike terms with variables, constants only combine with other constants. If you had another constant in the expression, you would add them together to simplify it.
To handle constants:

• Isolate or point them out in your expression.
• Add them together if there are multiple constants.
• Retain them as they are if alone, like the number 4 in this exercise.

Keep an eye on constants, as they simply add to or remain as part of the final simplified expression. For our example, the constant remains as 4.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. They are the backbone of algebra and help in solving many real-world problems.
An algebraic expression might seem complex initially because it contains:

• Variables like 'a', 'b', 'c' which represent unknown values.
• Constants or fixed numbers like 4 in our example.
• Coefficients, which are numbers multiplied by the variables (like 6 in \(6a\)).
• Operations such as addition, subtraction, multiplication, and division.

Understanding each component helps you simplify and manipulate these expressions. Breaking them down into manageable parts by identifying like terms and constants, as shown in the step-by-step solution, makes these expressions less daunting and more straightforward to work with.