Problem 75 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each algebraic expression. $$4+7 y-17 y$$

Step-by-Step Solution

Answer

The simplified expression is \(4 - 10y\).

Step 1: Identify like terms

In this problem, the like terms are \(7y\) and \(-17y\).

Step 2: Combine like terms

The subtraction of these like terms leaves \(-10y\). Therefore, the expression simplifies to \(4 - 10y\).

Key Concepts

Combining Like Terms

Understanding how to combine like terms is crucial in simplifying algebraic expressions. Like terms are terms in an expression that have the exact same variable part and raised to the same power. For instance, in the expression 4 + 7y - 17y, the terms 7y and -17y are like terms because they both contain the variable y to the first power. To combine them, you add or subtract their coefficients—the numerical part in front of the variable—while keeping the variable part the same.

When combining like terms, it's like adding or subtracting apples with apples; you handle the numbers and leave the item, in this case, the variable part, unchanged. The strategic pairing of terms that are alike makes simplification much more manageable. In our example, 7y and -17y combine to give -10y. Remember to keep an eye out for negative signs, as they indicate the direction of the operation. In this case, we're subtracting 17y from 7y.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (such as x, y, etc.), and arithmetic operations like addition, subtraction, multiplication, and division. An essential part of understanding algebraic expressions is recognizing that they represent quantities that can vary—that's what the variables are there for! They can be simple, with just one term, or complex, with several terms.

To tackle algebraic expressions systematically, start by identifying individual terms. Then, determine which terms are like (as discussed previously) and combine them accordingly. Always check for simplification possibilities, like reducing fractions or simplifying square roots where applicable. Simplifying algebraic expressions is not just about crunching numbers; it engages critical thinking skills, as you must decide the most efficient order of operations to arrive at the simplest form of the expression.

Subtracting Like Terms

When subtracting like terms, you're dealing with a specific case of combining like terms. It comes up when the terms you want to combine both contain the same variable and are connected by a subtraction sign. It’s simple if you remember basic subtraction rules and apply them to the coefficients of these like terms.

In subtraction, the order matters. Consider our exercise example with the terms 7y and -17y. You subtract the second term from the first, which means you calculate 7 - (-17). The two minus signs effectively turn the operation into an addition, giving us 7 + 17, resulting in 24y—but this doesn't apply here. Instead, we are subtracting 17y from 7y, thus 7 - 17 gives us -10, so the like terms combine to -10y. Always pay close attention to negative signs and treat the operation between terms exactly as you would a normal subtraction.