Problem 35 Perform the operations. $$ (... [FREE SOLUTION]

91影视

Elementary Algebra

John Redden

$Math Studyset 91影视 Explanations$ Math

2011 Edition

Chapter 5: Problem 35

Perform the operations. $$ (7 x 2 y 2-3 x y+6)-(6 x 2 y 2+2 x y-1) $$

Short Answer

Expert verified

The simplified result is $ x^2y^2 - 5xy + 7 $.

Step by step solution

Distribute Negative Sign

First, distribute the negative sign across the second polynomial. This changes the signs of each term within the parenthesis:\[ (7x^2y^2 - 3xy + 6) - (6x^2y^2 + 2xy - 1) \]This becomes:\[ 7x^2y^2 - 3xy + 6 - 6x^2y^2 - 2xy + 1 \]

Combine Like Terms

Next, combine like terms. Like terms are those that have the same variables with the same powers:- For $x^2y^2$ terms: $(7x^2y^2 - 6x^2y^2) = x^2y^2$- For $xy$ terms: $(-3xy - 2xy) = -5xy$- Constant term: $(6 + 1) = 7$The expression simplifies to:\[ x^2y^2 - 5xy + 7 \]

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Key Concepts

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

Combining Like Terms

Combining like terms is a fundamental step in simplifying polynomial expressions. When dealing with algebraic expressions, like terms are those that have identical variables raised to the same power. For instance, in the expression

$7x^2y^2$
$6x^2y^2$

these terms are like terms because both have the variables $x^2$ and $y^2$.
To combine them, simply add or subtract their coefficients:

$(7x^2y^2 - 6x^2y^2) = x^2y^2$
For $xy$ terms: $(-3xy - 2xy) = -5xy$
Constant terms: $(6 + 1) = 7$

Make sure every term pairs with a like term before performing arithmetic operations. This simplifies the expression into a more manageable form.

Distributing the Negative Sign

Distributing the negative sign is a crucial step before combining terms in expressions that involve subtraction. When there is a subtraction of a group of terms, distributing the negative sign effectively changes the signs of all terms inside the parentheses. For the expression:

$(7x^2y^2 - 3xy + 6) - (6x^2y^2 + 2xy - 1)$

we distribute the minus sign to each term in the second set of parentheses to obtain:

$7x^2y^2 - 3xy + 6 - 6x^2y^2 - 2xy + 1$

This step ensures that each term's original intent is retained, converting subtraction into an addition problem, which simplifies combining the terms.

Understanding Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operators (such as plus or minus signs). It is the foundational building block in algebra. In the expression:

$7x^2y^2 - 3xy + 6$

$7x^2y^2$ represents a term with a coefficient of 7 and variables $x$ and $y$ raised to the power of 2.
$-3xy$ signifies a term that includes both variables with their powers 1.
The constant 6 is a term without a variable, representing a fixed value.

Understanding these components will help with operations like addition, subtraction, and simplification of expressions by recognizing which terms can be combined or altered.

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91影视

Perform the operations. $$ (7 x 2 y 2-3 x y+6)-(6 x 2 y 2+2 x y-1) $$

Short Answer

Step by step solution

Distribute Negative Sign

Combine Like Terms

Key Concepts

Combining Like Terms

Distributing the Negative Sign

Understanding Algebraic Expressions

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