Problem 59 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(7.2 x-5.1+2.3 x+5.1\)

Step-by-Step Solution

Answer

9.5x

Step 1: Combine like terms

Identify and group the like terms in the expression. In this case, the like terms are the terms with the variable x and the constant terms.

Step 2: Simplify the variable terms

Add the coefficients of the variable terms:\[7.2x + 2.3x = (7.2 + 2.3)x = 9.5x\]

Step 3: Simplify the constants

Add the constant terms:\[-5.1 + 5.1 = 0\]

Step 4: Write the simplified expression

Combine the simplified variable terms with the simplified constants to get the final expression: \[9.5x + 0 = 9.5x\]

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, one of the key steps is combining like terms. Like terms are terms that contain the same variable raised to the same power. For example, in the expression given in the exercise,

• 7.2x and 2.3x are like terms because they both have the variable x raised to the first power.
• -5.1 and 5.1 are like terms because they are both constant terms.

Combining like terms means adding or subtracting the coefficients of the like terms. This helps to simplify the expression.

Coefficients

A coefficient is a numerical factor that multiplies a variable in an algebraic term. In the expression 7.2x, 7.2 is the coefficient and x is the variable. Let's look at a few key points about coefficients:

• Coefficients can be positive or negative.
• They determine the number of times the variable is multiplied.

In our example, we combined the coefficients of like terms to simplify the expression. We added 7.2 and 2.3 (the coefficients of x) to get 9.5, allowing us to rewrite 7.2x + 2.3x as 9.5x.

Variable and Constant Terms

In algebra, terms in an expression can be either variable or constant terms. It is important to identify these to simplify the expression correctly.

• A variable term includes a variable (like x in 7.2x), while a constant term is a number without any variables.
• Variables can represent unknown values and can change, hence the name 'variable.'
• Constants are fixed values and do not change.

In the original expression, 7.2x and 2.3x are variable terms, while -5.1 and 5.1 are constants. By combining like terms, we isolated the variable term and the constant term, resulting in the simplified expression 9.5x.