Question

The expression \(7(x+3)-3(2 x-2)\) is equivalent to: A. \(x+1\) B. \(x+15\) C. \(x+19\) D. \(x+23\) E. \(x+27\)

Short answer

Question: Simplify the expression \[7(x+3)-3(2x-2)\] and select the equivalent option from A to E. A. \(7x+13\) B. \(x+9\) C. \(-13x-4\) D. \(13x+25\) E. \(x+27\) Answer: E. \(x+27\)

Step-by-step solution

Apply the distributive property

First, distribute 7 to both terms inside the first parenthesis and -3 to both terms inside the second parenthesis: \[7(x+3)-3(2x-2)=7x+21-6x+6\]

Combine like terms

Now, combine the like terms (the x terms and the constant terms) in the expression. \[7x-6x+21+6=x+27\] So the simplified expression is \(x+27\). The correct answer is: E. \(x+27\)

Key concepts

Combining Like Terms

When working with algebraic expressions, combining like terms is an essential step towards simplification. Like terms are terms that have the same variable parts raised to the same powers, which means they can be added or subtracted from each other. To spot like terms, look for terms that have the same letter variables and check if the exponents on those variables match as well. For instance:

• Terms with just numbers, known as constant terms, like 2 and 4 or -5 and 3.
• Terms with the same variable, for example, 3x and -2x or 7y and 5y.
• Terms with the same variables and exponents, such as 2x^2 and -6x^2, but not 2x^2 and 3x^3 as the exponents are different.

Let's illustrate combining like terms with the expression from the textbook exercise. We had the terms 7x and -6x which are like terms because they both contain the variable x raised to the first power. Similarly, 21 and 6 are like terms since they are both constants. Combining 7x and -6x gives us x, and adding 21 and 6 we get 27, ending up with the simple expression x + 27.

Simplifying Expressions

Simplifying algebraic expressions is critical as it makes them easier to understand, evaluate, or use in further algebraic operations. The process typically involves several steps:

• Distributing multiplication over addition or subtraction, as seen with the distributive property.
• Combining like terms, as mentioned previously.
• Evaluating any simple arithmetic to reduce terms to their simplest form.

Consider the exercise's solution where we first distribute, then combine like terms. After the distribution, the expression had several terms that we then combined to simplify it significantly. Remember, the goal is to have the fewest terms possible without changing the value of the original expression. The simplified expression can then be used to evaluate for specific variable values or as part of larger algebraic equations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. An expression becomes 'algebraic' when it incorporates at least one variable, which is a symbol (often a letter) that represents an unknown value. Simple examples include 2x + 3 or 5y - 7.

Algebraic expressions become more complex when involving exponents, parentheses, or several terms, but the principles of simplification remain the same. The distributive property, for example, allows you to multiply a single term across terms inside parenthesis, a crucial step when dealing with expressions like those in the initial exercise. By mastering the methods of distribution, combining like terms, and overall simplification, you can confidently tackle a wide array of algebraic expressions.