Problem 51 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(-5 y+3-1+5+y-7\)

Step-by-Step Solution

Answer

-4y

Step 1: Combine like terms involving 'y'

Identify and combine all the terms that have the variable 'y'. In this expression, these terms are -5y and y. Combine them:-5y + y = -4y

Step 2: Combine constant terms

Identify and combine all the constant terms (numbers without variables). In this expression, these terms are +3, -1, +5, and -7. Combine them:3 - 1 + 5 - 7 = 0

Step 3: Write the simplified expression

After combining the like terms, the simplified expression is the sum of the results from the previous steps. Thus, we have: -4y + 0 which simplifies to just -4y

Key Concepts

Combining Like Terms

When working with algebraic expressions, it's essential to first identify and combine like terms. Like terms are terms in an expression that have the same variable raised to the same power. For example, in the expression \text{-5y + y}, the like terms are \text{-5y and y}.
Combining like terms helps simplify expressions and makes them easier to work with.
In our given exercise, we combine \text{-5y} and \text{y} to get:
\[ -5y + y = -4y \]

Combining Constants

Constants are numbers without variables. To simplify an expression, we also need to combine these constants. In the given expression \text{-5y + 3 - 1 + 5 + y - 7}, the constants are \text{3, -1, 5}, and \text{-7}.
Let's add them step by step:
\[ 3 - 1 = 2 \]
\[ 2 + 5 = 7 \]
\[ 7 - 7 = 0 \]
So, after combining the constants, we get 0.

Algebraic Simplification

Algebraic simplification involves reducing an expression to its simplest form. After combining like terms and constants, we write the simplified form of the expression.
In our example, after combining like terms (\text{-5y} and \text{y}) and constants (\text{3, -1, 5}, and \text{-7}), we have:
\[ -4y + 0 = -4y \]
Simplifying expressions helps in solving equations and inequalities more efficiently. Always remember to perform operations systematically to avoid errors.