Problem 50 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$-5 c+10+8 c-3$$

Step-by-Step Solution

Answer

The simplified version of the expression -5 c+10+8 c-3 is \(3c + 7\).

Step 1: Identify like terms

The like terms are the terms that have the same variables, which are -5c and 8c in this case. The constants in the expression are 10 and -3.

Step 2: Combine the like terms

Combine the like terms together using the basic arithmethic operations. Add -5c and 8c to get the combined term with the variable \(c\), and add 10 and -3 to get the combined constant term. This result in the partial simplified expression: \(3c + 7\).

Step 3: Check The Final Form

Once the like terms are combined, the expression cannot be simplified any further since there are no more like terms left. Thus, the final simplified version of the expression is \(3c + 7\).

Key Concepts

Like Terms

Like terms are terms that have identical variable parts, including any exponents. Understanding like terms is crucial for simplifying expressions easily. When you see an expression, like

• \(-5c\)
• \(+ 8c\)

this means both terms are "like" because they both contain the variable \(c\). You can only combine terms that are "like" because their similar structure allows us to perform arithmetic on them.
Constant terms, such as numbers without a variable, are also considered like terms with each other. In this expression,

• \(10\)
• \(-3\)

are like terms too. Remember: like terms simplify calculations and make solving algebraic problems faster.

Basic Arithmetic Operations

Basic arithmetic operations include addition, subtraction, multiplication, and division. These are the fundamental tools you need for simplifying expressions. To combine terms efficiently:

• Addition: Bring two like terms together by adding. Example: \(-5c + 8c = 3c\)
• Subtraction: Separate terms by subtracting. Example: \(10 - 3 = 7\)

When dealing with expressions that have both positive and negative numbers, careful application of these operations simplifies the task.
These operations allow you to take a messy, complicated expression and turn it into a neat, simplified equation. Practice with these operations is key to becoming comfortable working with algebraic expressions.

Combining Like Terms

Combining like terms is the process of adding or subtracting terms with the same variables and exponents to simplify an expression. This is very important in algebra because it reduces complex expressions into simpler forms, making them easier to understand and solve. By combining like terms:

• \(-5c + 8c\) simplifies to \(3c\)
• \(10 - 3\) simplifies to \(7\)

This process involves using basic arithmetic operations to reduce expressions to their simplest form. For example, given the expression \(-5c + 10 + 8c - 3\), identifying and combining like terms gives you a cleaner, simplified expression as \(3c + 7\). Always check after combining to ensure further simplification isn't possible, which leads to the expression being in its final form.