Problem 52 - Algebra 2

Question

Simplify \(7 x+8 y+9 y-5 x\).

Step-by-Step Solution

Answer

The simplified expression is \(2x + 17y\).

Step 1: Identify Like Terms

In the expression \(7x + 8y + 9y - 5x\), like terms are those that have the same variable raised to the same power. Here, the terms \(7x\) and \(-5x\) are like terms, and \(8y\) and \(9y\) are like terms.

Step 2: Combine Like Terms for 'x'

Combine the like terms for \(x\): \(7x - 5x\). Subtract \(5\) from \(7\) to get \(2x\).

Step 3: Combine Like Terms for 'y'

Combine the like terms for \(y\): \(8y + 9y\). Add \(8\) and \(9\) to get \(17y\).

Step 4: Write the Simplified Expression

Combine the results of the previous steps to write the simplified expression: \(2x + 17y\).

Key Concepts

Understanding Like Terms

In algebra, the concept of like terms is essential for simplifying expressions. Like terms are terms that have the same variable parts. This means they have the same variables and the same exponents. It's crucial to recognize like terms, as it allows you to combine them efficiently.

Here are some key points to help identify like terms:

• Look for terms with the same variables. For instance, in the expression \(7x + 8y + 9y - 5x\), the terms \(7x\) and \(-5x\) are like terms because they both contain the variable \(x\).
• Ensure the exponents on the variables are the same. An expression like \(3x^2\) and \(5x\) are not like terms because the exponents on \(x\) are different.
• Remember that unlike terms cannot be combined. They must remain separate in the final simplified expression.

Combining Like Terms

Once like terms are identified, the next step is to combine them. This simplification step makes expressions shorter and easier to work with, which is very helpful in solving further algebra problems.

Combining like terms involves these simple steps:

• Add or subtract the coefficients of the like terms. These are the numerical parts of the terms. For example, \(7x - 5x\) simplifies to \(2x\), since \(7 - 5 = 2\).
• Ensure that the variable and its exponent do not change when combining terms. Only the coefficient should change. In \(8y + 9y\), the result is \(17y\), with the variable \(y\) staying the same.
• Write down the new, combined term along with any unlike terms. In our example, after combining \(x\) and \(y\) terms separately, the simplified expression becomes \(2x + 17y\).

Algebraic Simplification

Simplifying algebraic expressions is a foundational skill in algebra. It involves making expressions as compact and efficient as possible by combining like terms.

When you simplify an expression:

• You make the expression easier to interpret and use. For example, simplifying \(7x + 8y + 9y - 5x\) into \(2x + 17y\) makes it much neater and clears the way for solving equations or evaluating expressions.
• Focus on operational efficiency. Fewer terms mean fewer calculations are needed in further operations.
• Emphasize understanding over memorization. The skill of recognizing and combining like terms helps you bypass unnecessary steps in more complex problems.

A well-simplified expression lays the groundwork for successfully tackling more advanced algebraic concepts. Understanding and practicing these steps ensure that students become more comfortable with algebra over time.