Problem 81 - Introductory Algebra for College Students

Question

Simplify each algebraic expression. $$8 x+x$$

Step-by-Step Solution

Answer

The simplified form of the expression \(8x + x\) is \(9x\).

Step 1: Identify like terms

In the expression, \(8x + x\), '8x' and 'x' are like terms. Here, x can be considered as \(1x\). So, the expression becomes \(8x + 1x\).

Step 2: Combine like terms

We can add the coefficients of the like terms. For '8x' and '1x', add the coefficients 8 and 1 to get 9. So, \(8x + 1x = 9x\).

Key Concepts

Like Terms

When working with algebraic expressions, 'like terms' are terms that have the same variable raised to the same power. In simpler words, the variable part of the terms must be identical. For instance, in the expression \(8x + x\), both terms are considered like terms because they both contain the variable \(x\). It doesn't matter if the coefficients, the numbers in front of \(x\), are different; the key is that variables match. This concept is pivotal when you need to combine terms, as it's only possible to combine terms that are alike. Ensuring you accurately identify like terms helps simplify the expressions effectively.

Coefficients

In algebraic expressions, a 'coefficient' is the number that multiplies a variable. For example, in \(8x\), the coefficient is 8, indicating that the variable \(x\) is multiplied by 8. In our given expression \(8x + x\), we can also write the second term as \(1x\) to make its coefficient visible. Coefficients play a critical role when you are combining like terms, as it’s the coefficients that are added or subtracted and not the variables. Properly understanding coefficients is essential for accurate calculations and simplifying mathematics.

Simplifying Expressions

Simplifying expressions involves combining like terms and performing arithmetic operations to reduce the expression to its simplest form. Start by identifying the like terms—in our case, it's \(8x + x\). Recognize that \(x\) is the same as \(1x\), which allows us to see clearly that both terms can be combined. Add the coefficients 8 and 1 together, resulting in 9. Therefore, the simplified expression becomes \(9x\). Simplification makes mathematical expressions easier to work with and is a crucial step in solving algebraic problems. This method can be applied to more complex expressions by breaking them down into simpler parts.