91Ó°ÊÓ

The distributive property is a fundamental concept in prealgebra that helps simplify expressions and equations. It allows us to eliminate parentheses by multiplying a term outside the parentheses with each term inside the parentheses. This property can be expressed as \(a(b + c) = ab + ac\). Here, \(a\) multiplies both \(b\) and \(c\), distributing evenly across the terms.
In the provided exercise, the expression \(5(4y - 6)\) uses the distributive property to remove the parentheses.

• First, multiply the 5 by \(4y\), resulting in \(20y\).
• Then, multiply 5 by -6, giving -30.

After distribution, you get the expression \(20y - 30\). This step is crucial because it transforms a more complex equation into a simpler form that can be easily managed.

Solving linear equations involves finding the value of the variable that makes the equation true. A linear equation is straightforward because it typically involves variables raised to the first power. The goal is to isolate the variable on one side of the equation.
In our example, after applying the distributive property, we have:

• Original equation: \(21y = 20y - 30\)
• Rewritten equation without parentheses: \(21y = 20y - 30\)

The next step is to isolate \(y\) by moving terms around.First, focus on getting all \(y\) terms on one side and the constants on the other. To do this:

• Subtract \(20y\) from both sides, which simplifies to \(21y - 20y = -30\).

This results in a simplified equation that can be solved for \(y\), which, in this case, is \(y = -30\).

Collecting like terms is an essential step when simplifying equations because it makes complex expressions easier to manage. Like terms are terms in an expression that have the same variable raised to the same power. By combining them, you simplify the equation.
In the given exercise, once the distributive property has been applied, you have:

• The equation: \(21y = 20y - 30\)

Since \(21y\) and \(20y\) are like terms (because they both are multiplied by the same variable \(y\)), you can rewrite the equation to make solving simpler.
Subtract \(20y\) from both sides: \(21y - 20y = -30\).This simplifies to \(y = -30\). By focusing on collecting like terms, the equation reduces, making it obvious and straightforward to find the value of \(y\). Recognizing and combining like terms helps quicken and clarify the process of solving equations.