Problem 11 - Prealgebra

Question

Simplify each expression. $$6+(n+7)$$

Step-by-Step Solution

Answer

The simplified expression is \( n + 13 \).

Step 1: Identify Like Terms

In the expression, identify the numbers that can be combined. We have a '6' and a '+7' that can be combined since they are both constants.

Step 2: Combine Constants

Add the constants '6' and '7' together to simplify the expression. Calculate: \[ 6 + 7 = 13 \] This simplifies the expression to: \[ 13 + n \]

Step 3: Write the Simplified Expression

After combining the constants, we rewrite the expression by placing the variable term next to the sum, resulting in \[ n + 13 \]. This is the simplified form of the original expression.

Key Concepts

Like Terms in Algebra

In algebra, like terms are important because they simplify expressions. Like terms are terms in an expression that have the same variable raised to the same power. Only coefficients, the numbers multiplying the variables, differ between like terms. This means:

• Terms without variables are constants, which are also like terms if they are added or subtracted (e.g., 5 in "5 + x")
• Terms with the same variable and power, such as "2n" and "3n" in the expression "2n + 3n + 5"

By combining like terms, you reduce the number of terms in an expression, making it easier to work with. For instance, in the expression "n + n", you can combine the like terms to get "2n". It's like gathering like objects together to make them simpler to manage. This principle will be important as you solve more complex algebraic expressions.

Combining Constants

Combining constants refers to the process of adding or subtracting numerical values without variables. In algebraic expressions, these provide a straightforward way to simplify equations and expressions. Constants are numbers on their own, such as 6 or 7 in our expression.

In the context of our exercise, we saw the constants 6 and 7 combined:

• Since both numbers are constants, they can be added directly.
• Adding 6 and 7 gives us 13, streamlining the expression.

When you combine constants, remember that only numbers without variables are involved. This keeps the simplification step easy to manage. Always remember to recombine any constants before addressing the terms involving variables in any algebraic formula.

Understanding Algebraic Expressions

Algebraic expressions consist of terms that are combined through addition, subtraction, multiplication, or division. These expressions include numbers, variables, or both. They form the building block of algebra and are used to describe relationships or problem situations mathematically. Here’s what makes up an algebraic expression:

• Constants: Fixed numerical values without variables.
• Variables: Symbols that represent unknown values, like "n".
• Coefficients: Numbers that multiply the variables, such as the "2" in "2n".

In our example expression "6 + (n + 7)", the term inside the parenthesis combines with the constant outside. ")First, combine the constants to get the expression in a more manageable form. This means understanding how each part of the expression contributes to the whole, leading to a final simplified equation of "n + 13". This represents the essence of working with algebraic expressions—merging all parts into their simplest form for clarity and ease of use.