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When working with algebraic equations, you are handling mathematical statements that express the equality of two algebraic expressions. These equations typically involve variables, which are symbols that represent unknown values. In the exercise given, we have an equation, 6y + 3 = 7(y - 1), which shows an equality involving the variable y. The goal is to find the value of y that makes the equation true.

To approach such problems, students must understand the fundamental principles of algebra, such as the ability to manipulate equations using addition, subtraction, multiplication, and division while maintaining equality. By systematically applying these operations, you can solve for the unknown variable. For instance, in our exercise, multiplication is applied to expand brackets, and addition or subtraction is used to collect like terms. Remember, whatever operation you do to one side, always do the same to the other to keep the equation balanced!

The distributive property is a key tool in algebra that allows you to multiply a single term by each term within a parenthesis. For example, in the given equation 6y + 3 = 7(y - 1), the number 7 is distributed to both y and -1 within the parentheses. This yields 7y - 7.

Understanding this property is crucial because it helps simplify expressions and is often used to eliminate parentheses, making it easier to combine like terms. This step is demonstrated in the solution by transforming the original equation into one that is easier to solve. It is worth noting that the distributive property applies to subtraction as well, meaning you can distribute a negative number across terms inside parentheses.

Isolating variables is the process of arranging an algebraic equation so that the unknown variable stands alone on one side of the equality. This means you'll need to perform a sequence of operations to 'move' all other terms to the opposite side.

In the original equation from the exercise, after applying the distributive property and rearranging terms, we end with 3 = y - 7. To isolate y, you simply need to undo whatever operation is being performed on it. Here, because y is being subtracted by 7, we do the inverse operation—adding 7 to both sides resulting in y = 10. This technique is fundamental in algebra, and it demonstrates the power of inverse operations to simplify equations and find the values of variables.