Problem 29 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ 7(x+6) $$

Step-by-Step Solution

Answer

Question: Expand and simplify the expression: $7(x + 6)$ Answer: $7x + 42$

Step 1: Apply the distributive property

To expand the expression, we should multiply each term inside the parentheses by 7, using the distributive property: $$ 7(x + 6) = 7 \cdot x + 7 \cdot 6 $$

Step 2: Perform the multiplications

Now, we need to perform the multiplications: $$ 7 \cdot x + 7 \cdot 6 = 7x + 42 $$

Step 3: Combine like terms (if any)

In this case, there are no like terms to be combined, so the final expression is simply: $$ 7x + 42 $$

Key Concepts

Combining Like Terms

To simplify algebraic expressions, one common task is to combine like terms. Like terms are terms in an expression that have the same variable raised to the same power, although they may have different coefficients. For example, in the expression 3a + 4a, both terms are like terms because they both contain the variable a to the same power, which is 1 in this case.

When combining like terms, you simply add or subtract the coefficients and keep the variable part unchanged. Using the above example, 3a + 4a would become 7a. It's crucial to only combine terms that are like terms. For instance, you cannot combine a and a^2 because the powers of a are different. Similarly, you cannot combine 7x and 42 in the provided exercise because one is a variable term and the other is a constant term.

Algebraic Expression Multiplication

Multiplying an algebraic expression involves using the distributive property to multiply a single term by two or more terms within parentheses. The distributive property states that for any numbers a, b, and c, the expression a(b + c) is equivalent to ab + ac. This means you distribute the multiplication of a to each term inside the parenthesis separately.

In the given exercise, multiplying 7(x + 6) involves applying the distributive property. Here, 7 is multiplied by both x and 6, leading to 7x + 42. This step is essential whenever an expression includes a term being multiplied by a set of terms inside parentheses. Understanding this property is crucial to mastering the basics of multiplication in algebra.

Elementary Algebra

Elementary algebra is the branch of mathematics that deals with variables and constants, along with the operations that unite them—addition, subtraction, multiplication, and division. It provides foundational tools for almost every field of mathematics and applied sciences. It involves solving equations and simplifying expressions, which is pivotal for advancing mathematical understanding.

Elementary algebra introduces students to the concept of using symbols and letters to represent numbers and the rules for manipulating these symbols. This framework is crucial for developing the skills to turn real-world problems into solvable equations. Concepts such as the distributive property and combining like terms are part of elementary algebra, and they are recurrent themes in various mathematical problems, such as the one demonstrated in the exercise. Understanding these fundamental algebraic concepts allows students to progress to more complex topics within mathematics.