Problem 33 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ 5(a-6) $$

Step-by-Step Solution

Answer

Question: Simplify the expression 5(a - 6). Answer: 5a - 30

Step 1: Distribute the constant to both terms inside the parentheses

We will multiply 5 by both terms inside the parentheses, \(a\) and \(-6\). This is also known as applying the distributive property. $$ 5(a-6) = 5 \times a + 5 \times (-6) $$

Step 2: Perform the multiplications

Now, multiply 5 by \(a\) and 5 by \(-6\). $$ 5 \times a + 5 \times (-6) = 5a - 30 $$

Step 3: Combine like terms (if any)

There are no like terms in this expression. So, the simplified expression is: $$ 5a - 30 $$

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operators like addition, subtraction, multiplication, and division. They are fundamental in algebra because they allow us to describe mathematical relationships and perform calculations in a general way.
A simple example of an algebraic expression is \(3x + 2\), where \(3x\) means "3 times some value of \(x\)," and \(+2\) indicates we add 2 to that result.

• Variables are symbols that represent unknown values. In our example, \(a\) is a variable.
• Coefficients are numbers that multiply the variables, such as the number 5 in \(5a\).
• Constants are standalone numbers, such as \(-30\) in \(5a - 30\).

Algebraic expressions are the building blocks for equations and inequalities, helping us solve real-world problems by representing them in a form that can be calculated.

Like Terms

Like terms in algebra are terms that have the same variables raised to the same power. They can be combined because they essentially represent the same unknowns and thus, can be simplified. For example, \(2x\) and \(3x\) are like terms because they both have the variable \(x\) and no exponents involved.

• To combine like terms, simply add or subtract their coefficients. For example, \(2x + 3x\) becomes \(5x\).
• The terms \(5a\) and \(-30\) are not like terms because one involves the variable \(a\) and the other is a constant.

Identifying and combining like terms is crucial for simplifying expressions so that they are easy to work with. In our problem, there were no like terms, so \(5a - 30\) remains unchanged after simplification.

Multiplication in Algebra

Multiplication in algebra involves multiplying numbers, variables, or a combination of both. It's a way to simplify expressions, particularly those within parentheses, using the distributive property. This property allows us to break down more complex expressions into simpler ones by distributing a factor to terms inside parentheses.
In our exercise, by applying the distributive property, we multiply 5 by each term within the parentheses

• Multiplying constants by variables, like \(5 \times a\), results in \(5a\).
• Multiplying constants by constants, like \(5 \times (-6)\), which results in \(-30\).

The distributive property simplifies calculations by helping us work with expressions logically, ultimately allowing us to derive accurate results, such as the expression \(5a - 30\).