Problem 73 - Elementary and Intermediate Algebra

Question

Simplify by combining like terms. $$ \frac{3}{5} t+\frac{1}{5} t $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{4}{5} t \).

Step 1: Identify Like Terms

In the expression \( \frac{3}{5} t + \frac{1}{5} t \), both terms are like terms because they share the same variable \( t \). Like terms can be combined by adding or subtracting their coefficients.

Step 2: Add Coefficients of Like Terms

Add the coefficients of the like terms: \( \frac{3}{5} + \frac{1}{5} \). Since they have a common denominator, you can add the numerators directly: \( \frac{3 + 1}{5} = \frac{4}{5} \).

Step 3: Form the Simplified Expression

Combine the coefficient with the variable \( t \) to form the simplified expression: \( \frac{4}{5} t \).

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that involves reducing an expression to its simplest form. It can make calculations easier and more straightforward. In the example given, the expression \( \frac{3}{5} t + \frac{1}{5} t \) is simplified by combining like terms. Like terms are those that have the same variables raised to the same power, in this case, terms with the variable \( t \). When simplifying, all you do is focus on the coefficients. For instance:

• Add or subtract just the coefficients if they are like terms.
• Ensure coefficients are properly combined by addition or subtraction when applicable.
• Keep the variable part intact during the process.

This approach not only simplifies calculations but also helps in easily recognizing whether the expression can be further transformed or used in an equation.

Fractions in Algebra

Fractions are commonly used in algebra and might seem daunting at first. However, understanding how to work with them can greatly help in solving algebraic expressions. For the expression \( \frac{3}{5} t + \frac{1}{5} t \), both terms are fractions. Here's how you handle them:

• Both terms share a common denominator, \( 5 \), which makes it straightforward to add the fractions.
• Simply add the numerators (the numbers on top), keeping the common denominator the same. Thus, \( \frac{3}{5} + \frac{1}{5} \) becomes \( \frac{3+1}{5} = \frac{4}{5} \).
• This result is then multiplied by the variable \( t \).

By working with fractions in this way, one can simplify and deal with complex algebraic expressions easily and effectively. Remember, finding a common denominator or reducing fractions are key skills when handling fractions in algebra.

Algebraic Expressions

Algebraic expressions are composed of numbers, variables, and mathematical operations. They are the foundation of algebra and can range from simple to complex. The expression \( \frac{3}{5} t + \frac{1}{5} t \) provides an example of a very basic algebraic expression. Let's break it down:

• The expression consists of terms where \( t \) represents a variable; in this context, it could stand for any number.
• Terms are combined using arithmetic operations such as addition and subtraction.
• Simplifying expressions means reducing them to a form that presents the least number of terms, in this case, combining like terms as seen above.

Understanding algebraic expressions involves recognizing these components and the operations used to combine them. Grasping these basics aids in tackling more advanced algebra problems as you progress in your studies.