Problem 48 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(19 x+x\)

Step-by-Step Solution

Answer

20x

Step 1: Identify Like Terms

Look at the expression and identify the like terms. In this case, both terms contain the variable x.

Step 2: Combine Like Terms

Combine the like terms by adding their coefficients. The coefficients are 19 and 1.

Step 3: Write the Simplified Expression

Add the coefficients together: 19 + 1 = 20. Therefore, the simplified expression is 20x.

Key Concepts

Combining Like Terms

Understanding how to combine like terms is a fundamental skill in algebra. Like terms are terms that contain the same variables raised to the same power. This means that in the expression 19x + x, both terms are 'like terms' because they both contain the variable 'x'.

When combining like terms, you simply add or subtract the coefficients of those terms. The coefficient is the numerical factor in front of the variable. Taking the given example, the coefficients of the terms 19x and x are 19 and 1, respectively.

By adding these coefficients together, we combine the terms into a single term with the same variable. So, combining 19x and x results in 20x.

Coefficients

Coefficients play a crucial role in algebraic expressions. They are the numbers that multiply the variables. In the expression 19x + x, 19 is the coefficient of the first term, and 1 is the coefficient of the second term (since x is the same as 1x).

To understand coefficients, think of them as the quantity of the variable they are attached to. In 19x, the coefficient 19 tells us that we have 19 instances of x. Similarly, in the term x, the implicit coefficient of 1 means we have 1 instance of x.

When solving algebraic expressions, your task is often to combine these coefficients correctly. In our example, adding 19 and 1 gives us 20.

Simplified Expression

A simplified expression is an algebraic expression that has been reduced to its most basic form. This means all like terms have been combined, and no further simplification is possible.

Simplifying an expression not only makes it easier to understand but also makes it more manageable for further calculations or solving equations.

When we simplified 19x + x to 20x, we combined the like terms to create a single term. The new expression 20x is much simpler and easier to work with than the original. This process of combining like terms and reducing the expression is essential for solving more complex algebraic problems.

Remember, a simplified expression will have no like terms left uncombined, and each term will be in its simplest form. This makes your algebraic expressions clearer and more concise.