Problem 12 - Prealgebra

Question

Simplify each expression. $$(3+k)+8$$

Step-by-Step Solution

Answer

The simplified expression is \(k + 11\).

Step 1: Identify like terms

In the expression \((3 + k) + 8\), identify the terms that are numbers and any variable terms. Here, 3 and 8 are numbers or constants, and \(k\) is a variable term.

Step 2: Simplify constants

Add the constant terms in the expression. Calculate \(3 + 8\), which equals 11. This gives us \(11 + k\).

Step 3: Write the simplified expression

Combine the terms to give the simplified expression: \(k + 11\).

Key Concepts

Like Terms

In algebra, when we talk about 'like terms', we refer to terms within an expression that have identical variable parts. Understanding this is crucial for simplifying expressions effectively.

A like term can include:

• Only constants, which are numbers without any variables.
• Variables raised to the same power and multiplied by any constant coefficients.

When simplifying an expression, you can only combine like terms. For example, in the expression \(3x + 4 + 5x + 7\):

• \(3x\) and \(5x\) are like terms because they both contain the variable \(x\).
• \(4\) and \(7\) are like terms because they are constants without variables.

Combining like terms helps in reducing the expression to its simplest form, making it easier to understand and work with.

Constants

Constants are the numbers in an expression that do not change. They are the fixed values without any variables attached. In straightforward terms, they are just regular numbers.

For instance, in the expression \(3 + k + 8\):

• \(3\) and \(8\) are constants because they have no variables that change their value.

When simplifying an expression:

• Gather and sum all constant terms for simplification.
• In this exercise, you add \(3\) and \(8\), which equal \(11\).

Combining constants allows you to reduce an expression and make solving equations more straightforward.

Variables

Variables are symbols used to represent unknown or changeable numbers in an expression or equation. The most common letters used for variables are \(x\), \(y\), \(z\), but any letter can serve as a variable. They are essential to forming algebraic expressions.

In the expression \(3 + k + 8\):

• \(k\) is the variable. It stands for an unknown number that can change depending on the problem's context.

Understanding variables helps in:

• Solving equations where you need to find the value of the variable.
• Representing real-life problems and relationships mathematically.

Variables are crucial as they allow us to create formulas and equations to model and solve various mathematical scenarios.