Question

Combine as indicated and simplify. $$7 a+5 a-9 a-3 a+5 a$$

Short answer

The simplified form of the expression is \(5a\).

Step-by-step solution

Combine like terms

Identify and combine the terms that contain the same variable powers. In this case, all terms contain the variable 'a'. So, simply add or subtract their coefficients as indicated: \[7a + 5a - 9a - 3a + 5a = (7 + 5 - 9 - 3 + 5)a.\]

Simplify the equation

Add the coefficients together: \[7 + 5 - 9 - 3 + 5 = 5.\] Now, multiply this sum by the variable 'a': \[5a.\]

Key concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that represent a specific mathematical idea. The process of combining like terms, as seen in the exercise \(7a+5a-9a-3a+5a\), is essential for simplifying these expressions.

When working with algebraic expressions, it is important to understand that terms are 'like' if they contain the same variables raised to the same power. For instance, \(7a\) and \(5a\) are like terms because they both contain the variable \(a\) raised to the first power. However, \(7a^2\) and \(5a\) would not be like terms, because their variables are raised to different powers.

To combine like terms effectively, group the coefficients (the numerical parts) of these like terms together, perform the required mathematical operations, usually addition or subtraction, and then rewrite the expression with the simplified term times the common variable.

Simplifying Equations

Simplifying an equation means transforming it into a simpler or more manageable form without changing its solution or the information it conveys. The aim is often to express the equation in terms of the fewest possible terms or to isolate the variable.

In the context of the example provided, simplification involves reducing the equation \(7a+5a-9a-3a+5a\) by combining like terms and then performing the arithmetic operations on the coefficients. This process turned the original longer expression into the much simpler \(5a\), which is easier to understand and work with in further calculations or applications.

Simplification is a foundational skill in algebra that enables students to solve more complex problems by breaking them down into more manageable parts, and it's often one of the first steps in the problem-solving process.

Mathematical Operations

Mathematical operations such as addition, subtraction, multiplication, and division are used to process numbers and variables in algebraic expressions. They follow a specific order of operations, commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

In the exercise \(7a+5a-9a-3a+5a\), the operations involved are addition and subtraction of the coefficients. Once you have identified the like terms, you then proceed with these operations. It's crucial to accurately apply these operations to simplify the expression correctly.

Understanding mathematical operations and their properties, like commutativity (order doesn't matter in addition or multiplication) and associativity (grouping doesn't matter in addition or multiplication), helps make the process of simplifying expressions more intuitive.