Problem 57 - Algebra 1: Concepts and Skills

Question

SIMPLIFYING EXPRESSIONS Simplify the expression. (Lesson 2.7) $$ 15-8 x+12 $$

Step-by-Step Solution

Answer

The simplified expression is \(27 - 8x\).

Step 1: Identifying the Elements in the Expression

First, identify the elements in the expression. Here, we have \(15\), which is a constant, \(-8x\), which is a term involving a variable, and \(12\), another constant.

Step 2: Grouping Constants

Next, group the constants together and combine them. In this case, we have \(15 + 12\), so we add those together like so: \(15 + 12 = 27\).

Step 3: Combining the Constants with the Variable term

Finally, we combine the result from step 2, which is \(27\), with the variable term, \(-8x\). Here, using the order of operations, subtraction comes after addition. Hence, we represent this equation as: \(27 - 8x\).

Key Concepts

Order of Operations

Understanding the order of operations is crucial when simplifying algebraic expressions to ensure calculations are performed correctly. The commonly followed sequence is remembered by the acronym PEMDAS - Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

In our example, there are no parentheses, exponents, multiplication, or division to consider, only addition and subtraction. According to PEMDAS, addition does not necessarily come before subtraction; rather, we solve operations from left to right. Therefore, when we simplify the expression \(15 - 8x + 12\), we start by combining the constants, as they are the first thing we encounter from left to right.

Combining Like Terms

Combining like terms is a fundamental aspect of simplifying algebraic expressions. Like terms are terms that have the same variables raised to the same powers. Constants, or numbers without variables, are also considered like terms with each other.

In the expression \(15 - 8x + 12\), the like terms are the constants 15 and 12. We combine them by addition or subtraction as appropriate. Since they're both positive numbers, we simply add them to get \(15 + 12 = 27\). We can't combine \(-8x\) with the constants because it contains a variable and is not 'like' the constant terms.

Variable Terms in Algebra

Variable terms in algebra are terms that include a variable, which is commonly represented by letters like \(x\), \(y\), and can have coefficients which are the numbers multiplying the variable.

In the given expression, \(-8x\) is a variable term with '-8' as its coefficient. Variable terms only combine with like variable terms, that is, terms with the same variable and exponent. The absence of other \(x\) terms in our example means \(-8x\) remains as it is when we simplify the expression.

Algebraic Expression Simplification

Simplifying algebraic expressions involves reducing them to their simplest form while ensuring the expressions remain equivalent to the original. This includes applying the order of operations, combining like terms, and understanding how to handle variable terms.

Bearing in mind the given solution steps, we first grouped and combined the constants (15 and 12) and then wrote down the simplified expression placing the single variable term last: \(27 - 8x\). This is the simplified form of the original expression. It's important to note that we do not change the sign in front of the variable term when combining; if it's negative, it stays negative, and the order relative to the constants is preserved from the original expression.