Problem 85 - Algebra A Combined Function

Question

Match each expression on the left with its simplification on the right. Not all letters on the right must be used and a letter may be used more than once. a. \(3 y\) b. \(9 y-6 y^{2}\) c. \(10 x\) d. \(25 x^{2}\) e. \(10 x-6\) f. none of these $$ (5 x-3)+(5 x-3) $$

Step-by-Step Solution

Answer

Match option (5x-3)+(5x-3) with 10x-6 (e).

Step 1: Understand the Problem

We need to match each expression on the left side (a to f) with its possible simplification or structure from the given options on the right.

Step 2: Review Given Expressions

The given expressions on the left are:* \(3y\)* \(9y - 6y^2\)* \(10x\)* \(25x^2\)* \(10x - 6\)* none of these.

Step 3: Analyze the Right Side Expression

The expression provided on the right is:\[(5x-3) + (5x-3)\].This expression needs to be simplified to see if it matches any of the left-side expressions.

Step 4: Simplify the Right Side Expression

Simplifying the expression \((5x-3) + (5x-3)\) involves combining like terms:\[5x - 3 + 5x - 3 = 10x - 6\].The simplified form is \(10x - 6\).

Step 5: Match Simplified Expression with Options

Compare the simplified expression \(10x - 6\) to the expressions on the left:* \(10x - 6\) matches exactly with the expression labeled as "e".

Step 6: Conclude the Solution

The simplified expression \((5x-3) + (5x-3)\) corresponds to the expression \(10x - 6\) which is labeled as "e" on the left side.

Key Concepts

Understanding Like Terms

In algebra, one of the fundamental concepts is knowing how to identify and combine like terms. But what exactly are "like terms"? Simply put, like terms are terms in an algebraic expression that have the same variables raised to the same powers. For example, in the expression \(3y + 5y\), both terms are like terms because they have the same variable, \(y\), and both have it raised to the first power (although it's often not written explicitly).Like terms can be easily added or subtracted. When simplifying expressions, such as \(5x - 3 + 5x - 3\), you look for these terms to combine them, which makes the expression easier to work with. In this case, the like terms would be \(5x\) and \(5x\); they combine to form \(10x\). Similarly, both \(-3\) terms are like terms (constant terms), so together they make \(-6\). Recognizing and combining like terms is crucial for simplifying expressions efficiently and accurately.

Exploring Polynomial Expressions

Polynomial expressions are a type of algebraic expression that includes variables and coefficients, involving operations such as addition, subtraction, and multiplication, but never division by a variable. These expressions are often classified based on the number of terms they have:

• Monomials: A single term, such as \(3y\) or \(25x^2\).
• Binomials: Two terms, such as \(10x - 6\).
• Trinomials: Three terms.

Each term in a polynomial comes together in a sum. For instance, the expression given in the step-by-step solution, \((5x-3) + (5x-3)\), simplifies to a binomial \(10x - 6\) by adding like terms. Polynomials can represent a wide range of numerical relationships and are used extensively in calculus and various branches of mathematics.Understanding the form and structure of polynomial expressions can greatly aid anyone solving algebra problems.

Algebraic Expressions: The Basics

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. Understanding them is essential in algebraic problem-solving, as they lay the groundwork for much of mathematics. The expression could be as simple as a single variable, like \(3y\), or it could be a more complex polynomial.The expressions behind the exercise are typical algebraic expressions:

• Each involves variables like \(x\) and \(y\).
• They include operations: addition \((+)\), subtraction \((-),\) and multiplication with the coefficients, such as \(9y - 6y^2\).

When you simplify these expressions, you are essentially making them easier to understand and work with. By knowing how to break these down into simpler components quickly, using steps like combining like terms, you become adept at tackling more complex equations.Grasping these basics of algebraic expressions helps form a solid foundation for further study in mathematics.