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Multiplication of integers is a fundamental operation in mathematics where we combine integer values to find their product. It's essential to know how to multiply integers correctly, especially when negative numbers are involved. When multiplying integers:

• If both integers are positive, the product is positive.
• If both integers are negative, the product is positive.
• If one integer is positive and the other is negative, the product is negative.

In the context of our exercise, we multiply \(-5\) by \(-y\). Since both numbers have negative signs, their product becomes a positive \(5y\). Similarly, when \(-5\) is multiplied by \(-7\), we apply the same rule, yielding a positive \(35\). Understanding these rules helps simplify expressions through multiplication, even when they appear complex.

Understanding positive and negative numbers is crucial in mathematics, as it helps in performing operations like addition, subtraction, and multiplication more efficiently. These numbers are represented on a number line:

• Positive numbers are greater than zero and are found on the right side of the number line.
• Negative numbers are less than zero and appear on the left side of the number line.

When dealing with multiplication involving negative numbers, it's vital to recognize that multiplying two negative numbers results in a positive product. This concept might seem a bit counterintuitive initially, but it aligns well with the rules we follow: a negative times a negative is positive. This is evident in the calculation of \(-5\times -7\), yielding \(35\), and \(-5\times -y\) giving \(5y\). Mastery of this concept allows smoother handling of expressions in algebra.

To simplify expressions, combining like terms is a key algebraic technique. Like terms are those that contain the same variable raised to the same power. When these terms are present, they can be added or subtracted with one another. For example, in the expression \(5y + 35\), \(5y\) and \(35\) are not like terms because \(5y\) involves the variable \(y\), while \(35\) is a constant.
To effectively combine terms, constantly look for terms that share the same variables and exponents, allowing more straightforward manipulation and simplification of algebraic expressions. This step allows us to express the final result compactly and prepare it for further operations or evaluations as needed in algebra. It's a powerful simplification tool that refines expressions, making them easier to work with.