91Ó°ÊÓ

In algebra, variables are symbols used to represent unknown values or quantities. Common variables include letters like \(x\), \(y\), and \(r\). Variables allow us to generalize mathematical expressions and form equations that can be solved to find unknown quantities.

For example, in the given problem, both terms \(8r\) and \(-13r\) contain the variable \(r\). This helps us understand that the expressions are related to the same unknown quantity. Thus, recognizing variables is the first step in identifying like terms in algebra.

Exponents play a crucial role in determining whether terms are like terms or not. An exponent indicates how many times a number or variable is multiplied by itself. For instance, in the expression \(a^2\), the number \(2\) is the exponent and tells us to multiply \(a\) by itself once: \(a \times a\).

In the example problem, the terms are \(8r\) and \(-13r\). Both have the variable \(r\) raised to the power of 1, even though the exponent is not explicitly written. Because both exponents are the same, it shows that each term involves the same variable raised to the same power, signifying that these terms are like terms. This uniformity in exponents is key for combining terms in algebraic expressions.

Algebraic terms consist of variables and constants combined through multiplication or division. In an algebraic expression like \(3x + 4y - 5\), \(3x\), \(4y\), and \(-5\) are individual terms. Each term is identified by its components: numbers known as coefficients, variables, and their exponents.

In the problem \(8r\) and \(-13r\), the coefficients are \(8\) and \(-13\), and the variable in both terms is \(r\). Since both terms share the same variable raised to the same power, they can be considered like terms. Understanding and identifying like terms is essential because it allows you to simplify algebraic expressions, making calculations and problem-solving more manageable.