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When simplifying algebraic expressions, the distributive property is a powerful tool that allows us to multiply a single term by each of the terms inside a bracket. Think of distribution like delivering mail; the single term outside the bracket needs to be 'delivered' to each term inside.

For example, in the expression 7(3y - 5), we use distributive property to multiply 7 by both 3y and -5, giving us 21y - 35. Similarly, the expression 2(4y + 3) becomes 8y + 6 when we distribute the 2. This step is pivotal because it sets up the equation for further simplification, allowing us to combine like terms in the next phase.

Once you have used the distributive property to remove parentheses, the next step is to look out for 'like terms.' These are terms that have the exact same variable parts raised to the same power. In the example from our exercise, we have 'like terms' 21y and 8y, which are both coefficients multiplied by the variable 'y'.

By adding these coefficients together, we combine the like terms to get 29y. Don't forget to also combine the constant terms, -35 and +6 in this case, which gives us -29. Combining like terms is essential because it simplifies the expression into a more manageable form, making calculation and understanding the relationship between variables and numbers much clearer.

An algebraic expression represents a mathematical phrase that can contain numbers, variables, and operation symbols. Think of it as a sentence where the numbers are the nouns, the variables are the characteristics that can change, and the operation symbols are the verbs that describe what to do with the numbers and variables.

In the exercise, 7(3y - 5) + 2(4y + 3) is an algebraic expression. It includes variables (y), coefficients (7, 3, -5, 2, 4, and 3), and operation symbols (multiplication and addition). Unlike equations, algebraic expressions don't have an equal sign. They are not complete sentences but rather thoughts or parts of equations that can be simplified and manipulated to reveal something about the relationships in a problem. By fully understanding the structure of an algebraic expression, one can simplify and solve more complex problems that build on these fundamental principles.