91Ó°ÊÓ

Algebraic expressions are the cornerstone of algebra and are comprised of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. A typical algebraic expression includes terms, which are the distinct entities separated by the plus or minus sign.

For instance, in the exercise \( \frac{5}{3}(5y + 6) - 10 = 0 \), we observe an algebraic expression which integrates numbers, a variable (\( y \)), arithmetic operations, and parentheses. Here, the expression includes a term outside the parentheses which is -10 and terms inside the parentheses which are 5y and 6.

Understanding how to work with algebraic expressions is crucial as it forms the basis for forming and solving equations, which in this case, leads to finding the value of the variable, \((y)\).

Combining like terms is a simplification process used to reduce an algebraic expression to fewer terms, which makes it simpler to work with. Like terms are the terms within an expression that have identical variable parts, regardless of their coefficients.

In the solution process of our example, after distributing \( \frac{5}{3} \) over the parentheses, we obtain \( \frac{25}{3}y + \frac{30}{3} - 10 = 0 \). Here, \(-10\) and \(+\frac{30}{3})\) are like terms because they do not contain any variables. By combining these like terms (which in itself is a subtraction, as 10 is negative), we simplify the expression to \( \frac{25}{3}y = 0 \). This step is vital for solving for \((y)\) and is a technique frequently used in algebra to streamline expressions and equations.

The distributive property is a rule in algebra that allows us to remove parentheses by distributing a factor across terms within parentheses. Specifically, it states that \(a(b + c) = ab + ac\), for any numbers or expressions \(a\), \(b\), and \(c\).

In our exercise, the distributive property is applied in step 1, as the expression \( \frac{5}{3}(5y + 6)-10 \) requires distribution of the fraction \( \frac{5}{3} \) across \(5y\) and \(6\). This is executed by multiplying \( \frac{5}{3} \) with each term inside the parentheses: \( \frac{5}{3} \times 5y \) and \( \frac{5}{3} \times 6 \). As a result, the parentheses are effectively removed and the expression simplifies to \( \frac{25}{3} y + 10 - 10 = 0 \). Mastery of the distributive property is fundamental in simplifying algebraic expressions and solving linear equations.