Question

Combine as indicated and simplify. $$5 a+2 a$$

Short answer

The simplified form of the expression is '7a'.

Step-by-step solution

Identify Like Terms

Examine the expression and identify like terms. In this case, the like terms are the terms that contain the variable 'a', which are '5a' and '2a'.

Combine Like Terms

Add the coefficients of the like terms together while keeping the variable part unchanged. For our terms, add the coefficients: 5 and 2.

Write the Sum

After adding the coefficients, combine them with the variable to get the simplified expression of the like terms. The final simplified expression for '5a + 2a' is '7a'.

Key concepts

Combining Like Terms

When you encounter algebraic expressions, one of the fundamental techniques for simplifying them is combining like terms. Like terms in algebra are terms that contain the same variables, raised to the same power. For instance, in the expression 5a + 2a, both terms are considered like terms because they contain the variable a raised to the first power.

To combine like terms, you simply add or subtract the coefficients of these terms. Coefficients are the numerical part that multiplies the variable, so in this example, the coefficients are 5 and 2. You keep the variable part unchanged and combine only the coefficients to get a new coefficient: 5 + 2 = 7. Therefore, 5a + 2a simplifies to 7a. Remember, you can only combine like terms—terms that have the same variable component.

Coefficients in Algebra

Understanding coefficients is crucial in algebra as they are essentially the numbers that characterize the size or 'weight' of the term they belong to. The coefficient in the term 5a is 5. When combining like terms, it's the coefficients that get added together, not the variables.

Consider the expression 5a + 2a again. The numbers 5 and 2 are coefficients of the term a. When you are tasked with simplifying an expression, you'll typically combine the coefficients of like terms to find the resulting expression. In more complex cases, be mindful of negative and fractional coefficients as they follow the same arithmetic rules.

It's easy to overlook the coefficient 1, as it is often omitted; for example, a is actually 1a. Always remember that every algebraic term has a coefficient, even if it's not explicitly written.

Algebraic Expressions

At the heart of algebra lies the algebraic expressions. These expressions consist of variables, numbers, and operations such as addition and multiplication. An example of a simple algebraic expression is 5a + 2a, where a is the variable, and 5 and 2 are the coefficients of the terms containing a.

An algebraic expression can be simplified by combining like terms, which involves adding the coefficients of terms with the same variable. However, remember that terms with different variables or powers cannot be combined. For instance, you cannot combine 5a with 2b or 5a^2 because they are not like terms.

To master algebraic expressions, work on identifying like terms and practice combining them correctly. This is a skill that will be used time and again in all sorts of algebraic problems.