Problem 14 - Intermediate Algebra

Question

Simplify the algebraic expressions by combining similar terms. $$ 8 x y^{2}-5 x^{2} y+2 x y^{2}+7 x^{2} y $$

Step-by-Step Solution

Answer

The simplified expression is \(10xy^2 + 2x^2y\).

Step 1: Identify Similar Terms

Identify similar terms in the expression. Similar terms will have the same variables raised to the same power. Here, we identify terms involving \(xy^2\) and terms involving \(x^2y\).

Step 2: Group Similar Terms

Group the terms that have identified similar types together: \((8xy^2 + 2xy^2)\) and \((-5x^2y + 7x^2y)\).

Step 3: Combine Like Terms

Combine the terms by adding or subtracting their coefficients. For \(xy^2\), add \(8 + 2\), resulting in \(10xy^2\). For \(x^2y\), add \(-5 + 7\), resulting in \(2x^2y\).

Step 4: Write the Final Expression

Combine the terms obtained from the previous step to write the simplified expression: \(10xy^2 + 2x^2y\).

Key Concepts

Combining Like Terms

When you see an algebraic expression like \(8xy^2 - 5x^2y + 2xy^2 + 7x^2y\), it might appear complex at first. But simplifying it can make it much more manageable. One crucial step is combining like terms. Like terms are parts of the expression that involve the same variables raised to the same powers. In our exercise, terms like \(8xy^2\) and \(2xy^2\) are considered like terms because they share the same variable components \(xy^2\). The same goes for \(-5x^2y\) and \(7x^2y\), which share the variable component \(x^2y\).

Combining these terms means adding or subtracting their coefficients, which are the numerical parts that precede the variables. In this case:

• For \(xy^2\): Add the coefficients 8 and 2 together, resulting in \(10xy^2\).
• For \(x^2y\): Add the coefficients -5 and 7 together, resulting in \(2x^2y\).

Incorporating this understanding helps to simplify the whole expression efficiently.

Similar Terms

Similar terms in algebra involve a crucial concept of recognizing which parts of the expression can be combined. These are the terms that have identical variable components and powers. In our expression \(8xy^2 - 5x^2y + 2xy^2 + 7x^2y\), identifying similar terms is the first effective step towards simplification.

• Look for terms that contain the exact same variables with the same exponents. Here, \(xy^2\) is present in both \(8xy^2\) and \(2xy^2\).
• Similarly, \(x^2y\) is the identical variable component found in both \(-5x^2y\) and \(7x^2y\).

By organizing terms this way, you set the stage for combining them, which greatly simplifies the expression. It transforms what appears to be a complex task into a well-structured approach to algebra.

Algebraic Simplification

Algebraic simplification emphasizes breaking down expressions into their simplest form. It involves understanding how each part of the expression contributes to its entirety and using basic algebraic operations to streamline it.

Begin the process by identifying groups of like and similar terms, as we've discussed. Algebraic simplification allows us to see the underlying pattern and structure in the expression \(8xy^2 - 5x^2y + 2xy^2 + 7x^2y\). Combining like terms to yield \(10xy^2 + 2x^2y\) is the heart of simplification, showing the beauty of clarity over initial complexity.

• Recognizing opportunities to combine and simplify terms reduces errors and makes manipulation of expressions cleaner and more intuitive.
• By practicing these techniques, students gain a better grasp of algebra as a whole and appreciate the elegance of mathematical problem-solving.

Ultimately, the goal is to make expressions easier to handle, which is beneficial not only in academic settings but also in problem-solving scenarios in various fields.