Problem 40 - Prealgebra

Question

Simplify expression. \(-2 y+x+3 y\)

Step-by-Step Solution

Answer

The simplified expression is \(y + x\).

Step 1: Identify Like Terms

In the given expression \[-2y + x + 3y\]you need to identify like terms. Like terms are terms that have the same variable raised to the same power. Here, \(-2y\) and \(3y\) are like terms because they both have the variable \(y\). The term \(x\) is different and stands alone.

Step 2: Combine Like Terms

Now, add the coefficients of the like terms together. For the like terms \(-2y\) and \(3y\), you add the coefficients \(-2\) and \(3\):\(-2 + 3 = 1\).So, \(-2y + 3y = 1y\) or simply \(y\).

Key Concepts

Like Terms

In algebra, like terms are expressions that have the same variables and are raised to the same power. Recognizing like terms is crucial when simplifying expressions because they can be combined to simplify calculations.

• Same Variables: Terms must include the same variable. For instance, both \( -2y \) and \( 3y \) contain the variable \( y \).
• Raised to the Same Power: The variables must have the same exponent. If \( y \) appears without an exponent, it's understood to be \( y^1 \).

When combining like terms, only the coefficients, the numerical part of the terms, are added or subtracted, not the variables. This helps to consolidate expressions into a simpler form that's easier to work with. In the example \( -2y + 3y\), since both terms include \( y \), you combine them by adding their coefficients, resulting in a simpler expression: \( y \).

Coefficients

Coefficients are numeric factors that multiply the variables in an algebraic expression. Understanding coefficients is crucial for simplifying expressions and solving equations.

• Role of Coefficients: They dictate how many times a term's variable is taken into account. In \( -2y \), \( -2 \) is the coefficient: it shows that the term involves \( y \) two times negatively.
• Combining Coefficients: When simplifying expressions, you add or subtract coefficients of like terms. For example, in the expression \( -2y + 3y \), add the coefficients \( -2 \) and \( 3 \) to get \( 1 \). Thus, \( -2y + 3y \) simplifies to \( 1y \) or simply \( y \).

Coefficients provide the magnitude and direction of the variable: positive coefficients add, while negative coefficients subtract. Grasping how coefficients function is foundational to mastering algebra.

Variables

Variables in algebra are symbols used to represent unknown or general values. They are the building blocks of algebraic expressions and equations.

• Purpose of Variables: They allow us to formulate general rules in mathematics and express relationships between quantities.
• Common Variables: Letters like \( x \), \( y \), and \( z \) are frequently used as variables.

Variables can take on different values, making expressions adaptable to various situations. When simplifying, the goal is to reduce the complexity of an expression while retaining the variable's symbolic nature. In the expression \( -2y + x + 3y \), both \( y \) and \( x \) are variables. Recognizing the variable helps identify which terms can be combined and which stand alone, like \( x \) in this case. Understanding variables is key to solving equations and modeling real-life scenarios.