Problem 87 - Introductory Algebra for College Students

Question

Simplify each algebraic expression. $$-y+4 y$$

Step-by-Step Solution

Answer

The simplified algebraic expression of \( -y + 4y \) is \( 3y \).

Step 1: Identify Like Terms

In this expression, \( -y \) and \( 4y \) are like terms because they both have the same variable, \( y \).

Step 2: Combine Like Terms

To simplify the expression, add the coefficients (the numbers attached to the variables) together.\nThis means you add \( -1 \) (the coefficient of \( -y \)) and \( 4 \) (the coefficient of \( 4y \)). This gives you \( -1 + 4 = 3 \).

Step 3: Write Final Expression

Since the coefficients add to 3 and the common variable is \( y \), the simplified expression is \( 3y \).

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, one of the first steps is to combine like terms. Like terms are terms in an expression that have identical variable parts, which means they have the same variable raised to the same power. To combine them, simply add or subtract the coefficients, which are the numerical parts in front of the variables. For example, in the expression \( -y + 4y \), \( -y \) and \( 4y \) are like terms since they both contain the variable \( y \).

Here's how you do it: Add the coefficient of \( -y \), which is implicitly \( -1 \) (since any variable without a coefficient has a coefficient of \( 1 \) or \( -1 \)), to the coefficient of \( 4y \) which is \( 4 \). The result is \( -1 + 4 = 3 \), and hence the combined term is \( 3y \). This process helps simplify the expression and lays the groundwork for solving more complex equations.

Algebraic Coefficients

Understanding algebraic coefficients is essential in simplifying algebraic expressions. A coefficient is a number that multiplies a variable. For instance, in the term \( 4y \), \( 4 \) is the coefficient, which suggests there are four \( y \) units. Coefficients provide the 'amount' of the variable, and without them, variables are considered to have a coefficient of \( 1 \).

It's important to note coefficients can be positive or negative. Negative coefficients, such as the \( -1 \) in \( -y \), indicate the opposite direction or a subtraction from a quantity. Algebra requires careful attention to these signs, as they influence the result when you are combining like terms.

Variable Manipulation

In algebra, variable manipulation is a crucial skill that involves rearranging and simplifying expressions or equations that contain variables. This can include operations such as combining like terms, distributing properties over addition or subtraction, and using inverse operations to solve for a variable. Effective variable manipulation allows for simplification of an expression into a more manageable or solvable form.

For example, manipulating variables often involves factoring or expanding expressions and properly applying the rules of arithmetic to combine or separate terms. Mastery of variable manipulation paves the way for tackling higher-level algebra and calculus.

Introductory Algebra

The journey into algebra begins with introductory algebra, which lays the foundation for understanding mathematical relationships using variables, coefficients, and constants. This level of algebra introduces the basic principles of solving for unknowns, simplifying expressions, and understanding equation functions.

It also encompasses the use of mathematical operations such as addition, subtraction, multiplication, and division in terms of variables and numbers. One of the most essential skills learned in this stage is how to properly combine like terms—a fundamental process that helps students delve into more complex algebraic concepts and problem-solving.