Problem 24 - Elementary Algebra

Question

Simplify. $$ y 2-3 y+5-y 2+9 $$

Step-by-Step Solution

Answer

The simplified expression is \(-3y + 14\).

Step 1: Distribute and Simplify

The expression given is slightly misleading due to the lack of multiplication signs. First, interpret it as: \( y^2 - 3y + 5 - y^2 + 9 \). Notice that \( y 2 \) refers to \( y^2 \).

Step 2: Combine Like Terms

Now, combine like terms: \[ y^2 - y^2 - 3y + 5 + 9 \] The terms \( y^2 \) and \(-y^2 \) cancel each other out.

Step 3: Simplify Further

With \( y^2 \) terms canceled, simplify the remaining terms: \[ -3y + 5 + 9 \] Combine the constant terms: \( 5 + 9 = 14 \).

Step 4: Final Simplified Expression

The final expression, after combining like terms and simplifying, is: \(-3y + 14\).

Key Concepts

Combining Like Terms in Polynomials

When working with polynomials, combining like terms is an essential technique. Like terms are terms in an expression that have identical variable parts raised to the same power. They can be combined by adding or subtracting the coefficients.

• First, identify like terms by looking for terms with the same variables and exponents.
• Next, add or subtract the coefficients of these like terms to combine them into a single term.

In the exercise, we started with the expression: \( y^2 - 3y + 5 - y^2 + 9 \).

• Recognize that \( y^2 \) and \( -y^2 \) are like terms because they have the same variable \( y \) raised to the power of 2. By combining them, they cancel each other, leaving us with no \( y^2 \) terms.
• The term \(-3y\) has no other like term, so it remains as it is.
• Finally, the numbers 5 and 9 are constant terms and can be added to form the single constant 14.

Simplification of Expressions

Simplification is the process of reducing an expression to its simplest form. This makes it easier to interpret and use in further calculations. Simplifying an expression involves:

• Combining like terms.
• Eliminating any unnecessary brackets or signs.

For example:- Start with \( y^2 - 3y + 5 - y^2 + 9 \).- After combining like terms, the expression becomes \(-3y + 5 + 9\).Next, simplify any arithmetic operations among the constants. In this case, adding 5 and 9 gives us 14. Consequently, the expression simplifies to \(-3y + 14\).Making an expression as simplified as possible helps avoid mistakes and makes further algebraic manipulations smoother.

Understanding Algebraic Expressions

Algebraic expressions are built using numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. It is crucial to grasp their structure to handle them proficiently in algebra.

• Expressions like \( y^2 - 3y + 5 - y^2 + 9 \) contain terms, which can include coefficients, variables, and exponents.
• The order of terms doesn't affect the expression's value but understanding how to organize and simplify them does.
• Each term is separated by a plus or minus sign.

When simplifying, consider:

• Arranging terms in a standard way, frequently starting with the highest powers of variables.
• Recognizing the roles of coefficients and constants within the expression.

Overall, understanding the elemental structure of algebraic expressions lays a strong foundation for accurately performing operations like factoring and solving equations.