Problem 75 - Elementary Algebra

Question

Simplify. (Assume all denominators are nonzero.) $$ y+x x 2-y 2 $$

Step-by-Step Solution

Answer

The simplified expression is \( 2x^2 - y \).

Step 1: Separate Terms

First, observe the expression and separate the terms for simplification. The expression given is \( y + x \times x \times 2 - y \times 2 \). Separate it into its individual terms as: \( y \), \( x \times x \times 2 \), and \( - y \times 2 \).

Step 2: Simplify the Multiplication

Simplify each multiplication term by performing the multiplication. For \( x \times x \times 2 \), calculate it step-by-step: \( x \times x = x^2 \) and then \( x^2 \times 2 = 2x^2 \). For \( -y \times 2 \), simply multiply the constants: \( -1 \times 2 = -2 \) resulting in \( -2y \).

Step 3: Write Simplified Expression

Combine the simplified terms to rewrite the expression. The terms \( y \), \( 2x^2 \), and \( -2y \) are now combined to form the expression: \( 2x^2 + y - 2y \).

Step 4: Combine Like Terms

Combine the like terms in the expression \( 2x^2 + y - 2y \). The terms \( y \) and \( -2y \) can be combined as they are like terms. Subtract: \( y - 2y = -y \).

Step 5: Final Expression

Substitute the simplified terms back and write the final expression. After combining like terms, the expression simplifies to \( 2x^2 - y \).

Key Concepts

Like Terms

Like terms are an essential concept in algebra simplification. They are terms that have the exact same variable raised to the same power. This similarity allows them to be added or subtracted from each other. For instance, in the expression we simplified, the terms \( y \) and \(-2y \) are like terms.
To identify like terms:

• Look for terms with identical variable parts. Here, both terms contain the variable \( y \) and have no exponents, making them like terms.
• The coefficients (numbers in front of the variables) do not need to be the same.

After identifying like terms, you can then combine them by performing the arithmetic operation—either addition or subtraction—as done with \( y - 2y = -y \). This process simplifies expressions and is crucial for finding the most concise form of algebraic expressions.

Multiplication Simplification

Simplifying multiplication in algebraic expressions involves multiplying constants, as well as multiplying variables.
In our exercise, we had the multiplication \( x \times x \times 2 \) which required simplification.
Here’s how to tackle such expressions:

• Start with the variables: Multiply \( x \times x \) to get \( x^2 \). This is because multiplying like variables is essentially adding their exponents. So, \( x^1 imes x^1 = x^{1+1} = x^2 \).
• Next, multiply the result by the constant: \( x^2 \times 2 = 2x^2 \). Here, we simply multiply the coefficient, which is \( 1 \) in front of \( x^2 \), with \( 2 \).

This results in the simplified term \( 2x^2 \). Simplifying multiplication accurately makes the entire expression more manageable, preparing it for further operations like combining terms.

Expression Rewriting

Expression rewriting involves adjusting the given algebraic expression while maintaining its equivalent value.
The goal is to present it in a simpler and more useful form. Let's break it down using the expression from our exercise.
Initially, we had: \( y + x \times x \times 2 - y \times 2 \)

• The first step in rewriting it was to handle multiplication and simplify, resulting in terms like \( 2x^2 \) and \(-2y \).
• Next, we rewrote the expression by rearranging to isolate and combine like terms: \( 2x^2 + y - 2y \).
• Finally, simplifying further by combining the like terms \( y \) and \(-2y\) yielded the rewritten expression \( 2x^2 - y \).

Rewriting expressions in this fashion allows you to clearly see each operation performed step-by-step, ensuring nothing is overlooked and that all terms are in their simplest form.