91Ó°ÊÓ

To simplify a complex algebraic expression, it is often helpful to use the distributive property, a handy algebraic rule. In essence, it allows you to multiply a single term outside the parentheses by each term inside. To frame it in a formula, think of it as follows: for any numbers or variables, \(a\times(b + c) = a\times b + a\times c\). Using the distributive property can substantially simplify expressions, especially when dealing with coefficients, like fractions. In our exercise \(\frac{1}{2}(-8 x+4 y)\), we apply this property to efficiently distribute \(\frac{1}{2}\) across both -8x and +4y. It's like sharing a piece of cake equally between friends; each term gets an even piece of the fractional coefficient. This methodical distribution is the critical first step to untangling the problem and paving the way toward a simplified expression.

Once the distributive property has been applied, the next step in simplifying algebraic expressions lies in executing algebraic operations. These operations include addition, subtraction, multiplication, and division, which we use to combine or simplify terms. In the context of our exercise, after distributing \(\frac{1}{2}\), we're left with basic multiplication: \(\frac{1}{2} * (-8x)\) and \(\frac{1}{2} * (4y)\).This is where the knowledge of negative numbers, multiplication, and working with fractions becomes important. For instance, multiplying a negative by a positive will always result in a negative. Understanding such foundational operations is crucial for correctly simplifying the expression to \( -4x + 2y\), our final answer.

Understanding fraction multiplication is vital when working with algebraic expressions containing fractions. To multiply a fraction by a whole number (or another fraction), multiply the numerators together to find the new numerator and then do the same with the denominators for the new denominator. However, when multiplying by a whole number, the denominator remains the same since a whole number can be thought of as itself over 1 (for example, \(4 = \frac{4}{1}\)).In our problem, \(\frac{1}{2}\) is multiplied by -8x and +4y, separately. To visualize this, you can imagine splitting each term into a fraction: \(\frac{-8x}{1}\) and \(\frac{4y}{1}\). By multiplying the numerators, we get \(\frac{1 * (-8x)}{2 * 1} = -\frac{8x}{2}\) and \(\frac{1 * 4y}{2 * 1} = \frac{4y}{2}\), resulting in \( -4x\) and \(+2y\) after simplification. This process pinpoints exactly how these multiplication rules apply in algebra, offering a clear path to the solution.