Problem 16 - Algebra 1: Concepts and Skills

Question

Identify the like terms in the expression. $$ 5 s^{2}-10 s^{2} $$

Step-by-Step Solution

Answer

The like terms in the expression \(5s^2 - 10s^2\) are \(5s^2\) and \(-10s^2\).

Step 1: Identify Terms

First, look at the expression \(5s^2 - 10s^2\). An expression can be split into terms, which are separated by addition or subtraction operations. In this case, the terms are \(5s^2\) and \(-10s^2\).

Step 2: Examine Variables and Powers

Next, examine the variables and their powers in each term. In both cases, the variable is \(s\) and the power is 2.

Step 3: Conclusion

Since both terms have the same variable raised to the same power, they are considered like terms. So, the like terms in the expression \(5s^2 - 10s^2\) are \(5s^2\) and \(-10s^2\).

Key Concepts

Identifying Like Terms

In algebraic expressions, identifying like terms is crucial for simplification and solving equations. Like terms are terms that have exactly the same variables raised to the same powers, even if their coefficients (the numbers in front of the variables) are different. For example, in the expression 5s^2 - 10s^2, both terms contain the variable s raised to the power of 2.

When combing like terms, add or subtract only the coefficients. Here, 5s^2 and -10s^2 are like terms because they have the same variable and power. So, you can combine them to simplify the expression to -5s^2, by subtracting the coefficients (5 - 10).

• Terms with different variables or powers, such as 2x and 3y, are not like terms.
• Terms with the same variable but different powers, such as x^2 and x^3, are also not like terms.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. Variables are symbols (usually letters) that represent unknown values. Expressions can vary greatly, ranging from simple ones like 2 + 3 to more complex ones like 4x - 7y + 2xy^2.

The beauty of algebra lies in working with these expressions to perform operations, simplify, and solve equations. For instance, the expression 5s^2 - 10s^2 illustrates how algebra can be like a puzzle, where combining like terms is a key step in finding a solution or simplifying the expression.

Variables and Powers

In algebra, variables are symbols that represent numbers and powers indicate the number of times a variable is multiplied by itself. A term like s^2 means s is being squared, or s * s. If you have different terms with the same variable and power, they can be simplified.

Understanding the roles of variables and powers is essential for operations involving algebraic expressions. In our example 5s^2 - 10s^2, both terms have the variable s raised to the second power. Hence, they can be easily combined because the base and the exponent are identical. This principle is one of the cornerstones of simplifying algebraic expressions.