Problem 2 - Algebra 1

Question

What are the like terms in the expression \(-6-3 x^{2}+3 y+7-4 y+9 x^{2} ?\)

Step-by-Step Solution

Answer

The like terms in the expression are ' \(-3 x^{2}\) and \(9 x^{2}\)', '\(3 y\) and \(-4 y\)' and '\(-6\) and \(7\)'.

Step 1: Identify the Terms

To start, identify all the terms in the expression, which are: \(-6\), \(-3 x^{2}\), \(3 y\), \(7\), \(-4 y\), and \(9 x^{2}\)

Step 2: Group the Like Terms

Next, group the like terms together. Like terms are ' \(-3 x^{2}\) and \(9 x^{2}\)', '\(3 y\) and \(-4 y\)' and the constants '\(-6\) and \(7\)'.

Step 3: Finish

So the like terms in the expression are ' \(-3 x^{2}\) and \(9 x^{2}\)', '\(3 y\) and \(-4 y\)' and '\(-6\) and \(7\)'.

Key Concepts

Combining Like Terms

When working with algebraic expressions, a fundamental skill is the ability to combine like terms. Like terms are terms that have the same variable raised to the same power. In simpler terms, they 'look' the same except potentially having different coefficients. For instance, in the expression ewline{-6-3 x^{2}+3 y+7-4 y+9 x^{2}}, the terms ewline{-3 x^{2}} and ewline{9 x^{2}} can be combined because they are both terms with the variable ewline{x} raised to the second power. Similarly, ewline{3 y} and ewline{-4 y} can be combined as they are both terms with the variable ewline{y} to the first power.

Combining like terms makes the expression simpler and more manageable. It's like organizing a room by putting similar items together; it becomes easier to see what you have and to work with the space. This process typically involves adding or subtracting the coefficients of the like terms. For example, to combine ewline{-3 x^{2}} and ewline{9 x^{2}}, you would add their coefficients (ewline{-3 + 9}), resulting in ewline{6 x^{2}}. Notice that the variable part remains unchanged.

Practical Tip on Identifying Like Terms

To accurately identify like terms, focus on the variable and its exponent. Ignore the coefficients at first and see if the variable parts match. Only if they are an exact match should you consider them like terms and proceed to combine them using arithmetic operations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that contain numbers, variables (like ewline{x} or ewline{y}), and operations (such as addition, subtraction, multiplication, and division). The expression ewline{-6-3 x^{2}+3 y+7-4 y+9 x^{2}} is an algebraic expression composed of several terms, where each term is a product of a coefficient and a variable to a power, or just a constant (a number on its own).

Breaking down the given expression into individual terms helps us to understand the structure and to simplify an expression. Terms can be separated by a plus (ewline{+}) or minus (ewline{-}) sign. For example, in our expression, ewline{-6} and ewline{7} are constants (terms with no variables), while ewline{-3 x^{2}}, ewline{3 y}, ewline{-4 y}, and ewline{9 x^{2}} are terms that involve variables.

Understanding the Function of Variables

Variables in algebraic expressions serve as placeholders for numbers and can represent unknown values we might solve for or specific values depending on the context. They’re what make algebra flexible and broadly applicable in solving a range of mathematical problems.

Simplifying Expressions

Simplifying expressions is a process of reducing an algebraic expression to its simplest form. This process typically involves combining like terms and performing any applicable arithmetic. For the expression ewline{-6-3 x^{2}+3 y+7-4 y+9 x^{2}}, simplifying the expression would mean first grouping and then combining the like terms, as shown in the step-by-step solution provided.

Once like terms are combined, the expression takes a more compact and simpler form. In our example, simplifying the expression would result in combining the ewline{x^{2}} terms and the ewline{y} terms, as well as the constants separately. After combining like terms, you would add or subtract the numerical coefficients to get the simplified algebraic expression.

Why Simplification Matters

Simplifying an expression makes it easier to work with, especially if you need to perform further operations or solve equations. It lays the groundwork for understanding and solving more complex algebraic equations and functions, highlighting the importance of mastering this foundational concept in algebra.