Problem 43 - Algebra 2

Question

Simplify. $$ \left(10 x^{2}-3 x y+4 y^{2}\right)-\left(3 x^{2}+5 x y\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(7x^2 - 8xy + 4y^2\).

Step 1: Distribute the Negative Sign

First, distribute the negative sign to each term inside the parentheses of the second polynomial. This changes the expression to:\[10x^2 - 3xy + 4y^2 - 3x^2 - 5xy\]

Step 2: Combine Like Terms

Identify and combine the like terms in the expression. Like terms are those terms that have the same variables raised to the same power.1. Combine the \(x^2\) terms: \[10x^2 - 3x^2 = 7x^2\]2. Combine the \(xy\) terms: \[-3xy - 5xy = -8xy\]3. The \(y^2\) term remains unchanged as there are no similar terms to combine: \[4y^2\]

Step 3: Write the Simplified Expression

Now, write the simplified expression by putting together all the combined like terms:\[7x^2 - 8xy + 4y^2\]

Key Concepts

Combining Like Terms

In algebra, an essential step in simplifying expressions is combining like terms. Like terms are those that share the same variables raised to identical powers.
For example, in the expression \(10x^2 - 3xy + 4y^2 - 3x^2 - 5xy\), the terms \(10x^2\) and \(-3x^2\) are like terms because they both have the variable \(x\) raised to the power of 2. Similarly, \(-3xy\) and \(-5xy\) are like terms because they both contain the product of different variables \(x\) and \(y\) multiplied together.

• Combine like terms to make expressions more manageable.
• Add or subtract coefficients of like terms while keeping the variable parts unchanged.
• Ensure that terms being combined truly have the exact same variables and powers.

Combining like terms simplifies the expression to a more concise form, making solving equations easier and more straightforward.

Distributive Property

The distributive property is a fundamental algebraic principle that helps in simplifying expressions and solving equations. It involves multiplying a single term and two or more terms inside a parenthesis. A negative sign in front of a parenthesis follows this rule to distribute itself to all terms inside:

• Given: \[a(b + c) = ab + ac\]
• Also applicable: \(-a(b + c) = -ab - ac\).

In the original exercise, we used the distributive property of subtraction:

• The expression \( -(3x^2 + 5xy) \) becomes \(-3x^2 - 5xy\).

This step is crucial because it changes the signs of all terms within the parentheses, allowing further operations like combining like terms.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. They form the basic building blocks for more complex math problems.
Consider the original expression \((10x^2 - 3xy + 4y^2)-(3x^2 + 5xy)\):

• Variables \(x\) and \(y\) represent unknown values.
• Exponents like \(x^2\) denote multiples of a variable.

Working with algebraic expressions involves several skills. It requires knowledge of math rules like the distributive property and the ability to recognize and combine like terms. Simplification, as done in the problem, reduces complex algebraic expressions to more manageable forms, helping in understanding and solving algebra-related problems more effectively.