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Understanding the distributive property is essential when solving equations that involve expressions in parentheses. It states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. In mathematical terms, if you have an equation like \(a(b + c)\), you can distribute the \(a\) across \(b\) and \(c\) to get \(ab + ac\).

For example, when faced with an equation such as \(-2(8 - y) = -6y\), the first step is to distribute the \(-2\) across the \(8 - y\). This means multiplying -2 by each term inside the parentheses: \(-2 \times 8\) and \(-2 \times -y\). As a result, the equation transitions to \(-16 + 2y = -6y\) after distribution. It's important to note that negative signs should be carefully handled during the distribution process to avoid common mistakes.

Once you've applied the distributive property, the next step is to combine like terms, which helps simplify the equation. Like terms are terms that have the same variables raised to the same power. In other words, they're terms that are 'alike' and can thus be added or subtracted from each other. When combining like terms, you adjust the coefficients (the numbers in front of the variables) while the variable part remains unchanged.

For instance, consider the equation \(-16 + 2y = -6y\); here the like terms are \(2y\) and \(-6y\). To combine them, you'll perform the operation indicated by their coefficients. Adding \(6y\) to both sides gives you \(-16 = -4y\). It's like gathering all the similar pieces together to see what you're working with more clearly. Remember, when like terms are on opposite sides of the equation, you must use an appropriate operation (in this case, addition) to move them together.

Isolating the variable is crucial for solving linear equations. It means manipulating the equation to have the variable of interest alone on one side of the equals sign and the constants on the other. This step gives you the solution to the equation: the value of the variable that makes the equation true.

In the context of our equation \(-16 = -4y\), the variable \(y\) is multiplied by -4. To 'free' \(y\), you perform the inverse operation of multiplication, which is division. By dividing both sides of the equation by -4, you get \(y = 4\). Ta-da! The variable is now isolated, and you have found its value. Always perform the same operations on both sides of the equation to maintain its balance; this is similar to ensuring both sides of a scale have equal weight to stay in equilibrium.