Problem 27 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 8 a x+2 a x+6 a x $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression: $$8ax + 2ax + 6ax$$ Answer: $$16ax$$

Step 1: Identify Like Terms

In the algebraic expression $$8ax + 2ax + 6ax$$, all three terms are like terms because they have the same variables: a and x.

Step 2: Combine Like Terms

To combine like terms, add the coefficients of each term together. For this expression, the coefficients are: - 8 for the first term (8ax) - 2 for the second term (2ax) - 6 for the third term (6ax) When we add these coefficients together, we get: $$8 + 2 + 6 = 16$$

Step 3: Write the Simplified Expression

Now that we have combined the coefficients, we can write the simplified expression: $$16ax$$ So the simplified algebraic expression is: $$8ax + 2ax + 6ax = 16ax$$

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, one of the crucial first steps is combining like terms. But what are like terms? They are terms that have exactly the same variable parts raised to the same power. For example, in the expression 8ax + 2ax + 6ax, all terms are considered like terms because they contain the same variables, a and x, with no exponents.

To combine them, we simply add or subtract their coefficients. In our exercise, we have the coefficients 8, 2, and 6. These can be added together: \(8 + 2 + 6 = 16\), which gives us a single term, \(16ax\), greatly simplifying the expression.

Algebraic Coefficients

An algebraic coefficient is a numerical factor that multiplies a variable within an algebraic term. In the expression \(8ax\), \(8\) is the coefficient, while \(a\) and \(x\) are the variables. Coefficients are important because they tell us how much of a variable is in play. Thinking about coefficients as multipliers allows us to combine like terms effectively by adding or subtracting these multipliers accordingly.

Understanding coefficients is key to simplifying expressions as it determines the degree of variables we're working with. Once coefficients are combined, the expression becomes more compact and easier to work with.

Elementary Algebra

Elementary algebra is the branch of mathematics that deals with variables and constants. It builds upon arithmetic by including letters known as variables to represent numbers, allowing for general statements about relationships. An essential aspect of elementary algebra is manipulating and simplifying expressions. Simplification may involve a series of methods such as combining like terms, solving equations, and factoring.

Our exercise is a prime example of using elementary algebra to simplify an algebraic expression by combining like terms. By practicing such problems, students learn to apply the rules of algebra to make calculations and solve problems more directly and efficiently.

Variables in Algebra

In algebra, variables are symbols used to represent unknown numbers or values. Commonly, letters from the end of the alphabet, such as \(x\), \(y\), and \(z\), are used for this purpose. Variables are one of the fundamental building blocks of algebraic expressions and equations. They can stand alone, as in \(x\) or \(y\), or be part of more complicated terms combined with coefficients, as in \(2x\) or \(7y\).

Understanding variables and how they interact with numbers (coefficients) and other variables is crucial in algebra. It allows us to generalize arithmetic to work with expressions and equations that can represent a multitude of possible values or situations, thereby expanding the power and applicability of mathematics.