Problem 62 Simplify each algebraic expressi... [FREE SOLUTION]

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Elementary Algebra

Jerome E. Kaufmann, Karen L. Schwitters

$Math Studyset 91影视 Explanations$ Math

9 Edition

Chapter 1: Problem 62

Simplify each algebraic expression by combining similar terms. $$-(y-2)-(y+4)-(y+7)-2(y+3)$$

Short Answer

Expert verified

The simplified expression is $-5y - 15$.

Step by step solution

Distribute the Negative Signs

Rewrite the expression while distributing the negative signs in front of the parentheses: \[ -(y-2) = -y + 2 \] \[ -(y+4) = -y - 4 \] \[ -(y+7) = -y - 7 \] Combine these results to get: \[ -y + 2 - y - 4 - y - 7 \] Additionally, distribute the $-2$ in $-2(y+3)$ to get:\[ -2y - 6 \] So the full expression is: \[ -y + 2 - y - 4 - y - 7 - 2y - 6 \]

Combine Like Terms

Gather all the like terms in the expression. Focus on combining the terms with $y$ and the constant terms: - For $y$-terms: - Combine $-y, -y, -y,$ and $-2y$ to get $-5y$.- For constant terms: - Combine $2, -4, -7,$ and $-6$ to get $-15$.So, the simplified expression is:\[ -5y - 15 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combining Like Terms

To simplify an algebraic expression, the first step is often to combine like terms. This means grouping together terms that have the same variables raised to the same power. In our expression, we have variables all involving the same variable,
which makes them all like terms.

Identify the terms with the same variable: In the given expression, $-y, -y, -y,\text{ and } -2y$
Combine them by adding their coefficients: Combine $-1y + (-1y) + (-1y) + (-2y)$ to get $-5y$

By combining them, you simplify your expression, which reduces potential calculation errors later on. Always double-check that you've grouped all similar variables together.

Distributing Negative Signs

In expressions with parentheses, distributing signs, especially negatives, is crucial. Negative signs before a parenthesis reverse the signs of each term inside, which affects how we proceed.

Take note of the negative signs: In our case, negative signs need to be applied to $y-2,$ $y+4,$ $y+7,$ and $y+3$
Distribute the negative across the parentheses: This transforms $y-2$ to $-y + 2,$ $y+4$ to $-y - 4,$ and so on

Always perform this operation before combining like terms, as it ensures all signs are correctly accounted for and the correct expression is simplified. Remember,
distribution ensures expression simplification flows seamlessly.

Simplifying Expressions

Once distribution and combination of like terms are complete, we reach the crucial stage of simplifying the overall expression. Simplification means condensing the expression into its simplest form, making it easier to interpret or apply further operations.

Check your expression: Ensure like terms are grouped, and subtraction/negative distribution is correctly applied
Reduce the expression: For example, grouping all $y$-terms and constants, as detailed earlier, provides $-5y - 15$

Simplifying is a core part of working with algebraic expressions, helping tidy them up for clearer understanding or further mathematical manipulation. Revisit any steps where the simplification might seem too fast-paced, ensuring you grasp each stage thoroughly before moving forward.

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91影视

Simplify each algebraic expression by combining similar terms. $$-(y-2)-(y+4)-(y+7)-2(y+3)$$

Short Answer

Step by step solution

Distribute the Negative Signs

Combine Like Terms

Key Concepts

Combining Like Terms

Distributing Negative Signs

Simplifying Expressions

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