Problem 3 - Algebra 1: Concepts and Skills

Question

Simplify the expression by combining like terms if possible. If not possible, write already simplified. $$5 r+r$$

Step-by-Step Solution

Answer

The simplified form of the expression \(5r + r\) is \(6r\).

Step 1: Identify Like Terms

In this expression, the like terms are \(5r\) and \(r\). Both have the variable \(r\) raised to the power of 1.

Step 2: Combine Like Terms

Now, as we know that adding like terms fundamentally means adding their coefficients, we have \(5r + 1r = 6r\). Note: if a variable has no numerical coefficient, it's understood to be 1, hence \(r = 1r\).

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions is an essential skill in algebra. When you encounter an expression, the goal is to make it as simple as possible by performing operations that reduce it, such as adding, subtracting, multiplying, or dividing terms.

• Start by identifying any like terms, which are terms that contain the same variables raised to the same power.
• Then combine those like terms by performing arithmetic operations on their coefficients.

This process helps in understanding the overall structure of equations and eases solving them later. In the given exercise, the expression \(5r + r\) can be simplified by combining like terms. Identifying and combining like terms is a key step to simplifying expressions, as it condenses the expression and makes it more manageable.

Like Terms

Like terms are components of an expression that have identical variables and exponents. Recognizing them is crucial when simplifying algebraic expressions.

• For instance, in the expression \(5r + r\), both terms contain the variable \(r\) to the power of 1.
• This means they are like terms and can be combined by adding their coefficients.
• Terms like \(3x^2\) and \(5x^2\) are also like terms because they share the same variable \(x^2\).

In the original expression, since 5r and r are like terms, we can combine them to make the expression more compact. Understanding what constitutes like terms allows you to efficiently perform algebraic manipulations.

Algebraic Coefficients

In any algebraic term, the coefficient is the numerical part that is multiplied by the variable. It tells you how many times the variable is being added together.

• For example, in the term \(5r\), the coefficient is 5.
• If a term includes a variable without an explicit numerical coefficient, it is understood that the coefficient is 1. Therefore, \(r = 1r\).

In our exercise, adding the coefficients of the like terms, which are 5 and 1, gives a new coefficient of 6. This results in the simplified expression \(6r\). Understanding coefficients allows you to better manipulate and simplify equations by seeing how variables interact with numbers.