Problem 47 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each algebraic expression. $$7 x+10 x$$

Step-by-Step Solution

Answer

The simplified algebraic expression is \( 17x \).

Step 1: Identify like terms

Like terms are terms with the same variable raised to the same power. In this exercise, \( 7x \) and \( 10x \) are like terms because they both contain the variable \( x \) raised to the power of 1.

Step 2: Combine like terms

To combine like terms, add the coefficients (the numbers in front of the variable) together. The variable part remains the same. In this case, add the coefficients 7 and 10 to get 17, so \( 7x + 10x = 17x \).

Key Concepts

Like Terms in Algebra

Understanding like terms in algebra is essential when you're simplifying algebraic expressions. Like terms are terms that have exactly the same variables raised to the same power. To identify them, just look for terms that share the same letter and exponent. For instance, in the terms 7x and 10x, both terms have the variable x raised to the first power, which usually isn't written out. So, what’s the takeaway?

If two terms have matching variables and exponents, they’re like terms and you can simplify the expression by combining them, which we’ll go deeper into in the next section.

Combining Like Terms

Once you've identified like terms in an algebraic expression, you can streamline the expression by combining them. This process involves adding or subtracting their coefficients—the numerical part in front of the variables. But here's the golden rule: only the coefficients change; the variable part stays the same. For example, in our exercise with 7x + 10x, we add the coefficients, 7 and 10, to end up with 17x. Think of it as grouping similar items; just how a basket of apples stays a basket of apples regardless of the number you add. This process is crucial because it simplifies expressions, making it much easier for us to work with them in equations or larger algebra problems.

Algebraic Coefficients

The term 'coefficient' in algebra refers to the number that multiplies a variable. It’s like the numerical sidekick to the variable superhero. In the expression 7x, 7 is the coefficient of the variable x. Coefficients determine the size and direction (when considering positive or negative signs) of the terms they belong to. In the process of combining like terms, coefficients are essentially what we're adding or subtracting. It’s important to accurately combine coefficients to maintain the integrity of the mathematical relationship they represent. Remember, while coefficients can be positive or negative, they are always numbers and never involve the variables when we’re combining like terms.