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Chapter 2: Problem 52

Simplify each expression. See Examples 3 through 7 . Subtract \(4+3 y\) from \(8-5 y\)

Short Answer

Expert verified
The simplified expression is \(4 - 8y\).

Step by step solution

01

Write the Expression to be Simplified

The expression requires subtracting the polynomial \(4 + 3y\) from \(8 - 5y\). The operation can be written as: \((8 - 5y) - (4 + 3y)\).
02

Distribute the Negative Sign

Since we are subtracting \( (4 + 3y) \), we must distribute the negative sign across both terms in the parentheses. The expression becomes \( 8 - 5y - 4 - 3y \).
03

Combine Like Terms

Now, combine the like terms. Start with the constants: \(8 - 4 = 4\). Then combine the y terms: \(-5y - 3y = -8y\).
04

Write the Simplified Expression

Combine these results into the simplified form: \(4 - 8y\).

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Key Concepts

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

Distribute Negative Sign
When simplifying expressions that involve subtraction, an important first step is distributing the negative sign. Consider the expression \((8 - 5y) - (4 + 3y)\). The minus sign in front of the parentheses changes the sign of each term within them. If it's helpful, imagine the negative sign as \(-1\) being multiplied by each term inside the parentheses.
For example:
  • Distribute the negative: \( -(4 + 3y) \) becomes \(-4 - 3y\).
Always be careful during this step, as missing a negative sign during distribution can lead to errors in your calculations. Each term inside the parentheses changes its sign.
  • If a term is positive, it becomes negative.
  • If a term is negative, it becomes positive.
Effectively managing this step sets a solid groundwork for accurately simplifying polynomial expressions.
Combine Like Terms
Once the negative sign has been distributed, the next step involves combining like terms. Like terms are terms in an expression that have the same variable raised to the same power. In the problem \(8 - 5y - 4 - 3y\), 'like terms' consist of constants and variable terms that can be added or subtracted.

Here's how you handle them:
  • Identify constant terms and add them: \(8 - 4 = 4\).
  • Identify variable terms: \(-5y\) and \(-3y\). Combine them by addition: \(-5y - 3y = -8y\).
It鈥檚 crucial to carefully combine terms to ease the process of simplifying expressions and prevent mistakes. Take your time identifying and operating on like terms to ensure the simplified expression is accurate.
Simplify Expressions
Simplifying expressions is the final step in our process, where we bring everything together. After distributing the negative sign and combining like terms, the expression \(8 - 5y - 4 - 3y\) reduces to \(4 - 8y\).
The goal of simplification is to express the polynomial in its simplest form. This means you should have the least possible number of terms, with no operations left to perform.
Why simplify?
  • Makes expressions easier to understand.
  • Simplifies further calculations and analysis.
To check your work, ensure that all like terms have been combined, and every negative sign has been properly distributed. Approaching each step systematically makes simplifying polynomial expressions manageable and efficient.

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