Problem 60 - Algebra A Combined Function

Question

Perform each indicated operation. Don't forget to simplify if possible. Add \(3 y-5\) to \(y+16\)

Step-by-Step Solution

Answer

The result of the addition is \(4y + 11\).

Step 1: Write the Expression

Start by writing the given expressions to be added together. The expressions are \(3y - 5\) and \(y + 16\).

Step 2: Combine Like Terms

When adding polynomial expressions like these, combine the like terms. The like terms are the terms that have the same variable raised to the same power. For these two expressions: - Combine the \(y\) terms: \(3y + y\)- Combine the constant terms: \(-5 + 16\)

Step 3: Simplify

Perform the arithmetic in each group of like terms:- \(3y + y = 4y\)- \(-5 + 16 = 11\)Thus, the simplified expression is \(4y + 11\).

Key Concepts

Combining Like Terms

In algebra, a crucial step in solving equations involves combining like terms. This means that we group together and simplify parts of an expression where the terms share the same variable raised to the same power. For instance, in the expression \(3y + y\), both terms include the same variable \(y\). By combining them, we add their coefficients together to get \(4y\).
This process helps in reducing complexity and organizing the expression. It allows us to transform a cluttered equation into a more comprehensible, streamlined form. This technique is fundamental in algebra and gets frequently used when dealing with polynomial operations like addition, subtraction, and sometimes multiplication. Remember that only the terms with the same variable components can be combined, while different variables or powers remain separate.

Simplifying Expressions

Simplifying expressions refers to the step-by-step process where we make an algebraic expression as efficient and straightforward as possible. This involves combining like terms, as explained before, along with executing any simple arithmetic operations. In the original problem, simplifying the combined expression \(3y - 5 + y + 16\) meant performing the operation \(-5 + 16\) to arrive at 11, and adding the like terms \(3y + y\) to get \(4y\).
These simplifications are essential because they make the expressions easier to work with, particularly when solving equations or evaluating algebra problems. A simplified expression can be handled more effectively and allows us to pinpoint solutions to equations much faster, creating a more intuitive process for solving complex mathematical relationships.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. Variables represent unknown values and are typically denoted by symbols like \(x\), \(y\), or \(z\).
In the exercise, expressions like \(3y - 5\) and \(y + 16\) are algebraic expressions. They can contain coefficients (the numbers multiplying variables), constants, and variables themselves. The ability to manipulate these expressions through operations like addition and simplification is foundational in algebra. Once you master these basics, solving equations and understanding polynomials becomes much simpler. Algebraic expressions are everywhere—from describing physical phenomena to calculations in everyday life—making them an invaluable tool in both academic and real-world contexts.