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Division is a mathematical operation where you split a number (called the dividend) into equal parts. You use another number (called the divisor) to determine how many parts you'll have. When you perform a division, you get the result known as the quotient. The basic idea is to see how many times the divisor fits into the dividend.

For example, if you have a dividend of 20 and a divisor of 4, the division \( 20 \div 4 \) yields a quotient of 5. This means that 4 can enter into 20 exactly five times. In this particular problem, dividing \(-63\) by \(-7\) gives us 9 because negative numbers divided by negative numbers result in positive outcomes.

Understanding division can simplify problems and make calculations easier. It is especially helpful when dealing with fractions, percentages, and other complex arithmetic operations.

Negative numbers are numbers that are less than zero. They often represent values like debts, temperature below freezing, or just amounts below a set reference point. On a number line, negative numbers lie to the left of zero. Working with negative values requires following specific rules, especially when they interact with other numbers.

• Adding a negative number is like subtracting its positive counterpart (e.g., \(-5 + (-3) = -8\)).
• Subtracting a negative number turns into addition (e.g., \(5 - (-3) = 5 + 3 = 8\)).
• Multiplying two negative numbers results in a positive product (e.g., \((-2) \times (-3) = 6\)).
• Similarly, dividing two negative numbers yields a positive quotient (e.g., \(-63 \div -7 = 9\)).

Using these rules will help in simplifying expressions and understanding the results of mathematical operations involving negative numbers.

Substitution is a technique used in mathematics where a specific value replaces a variable in an expression or equation. This helps in solving expressions by making them simpler or by helping you find a direct numerical answer.

In this exercise, substitution is used by replacing the variables \(c\) and \(d\) with \(-63\) and \(-7\), respectively. This process transforms the algebraic expression \( \frac{c}{d} \) into a straightforward numerical division \( \frac{-63}{-7} \).

• To substitute, know which values will replace the variables.
• Re-write the expression with these values substituted in place of the variables.
• The expression can now be evaluated using standard arithmetic operations.

By using substitution effectively, challenging algebraic expressions become manageable and transparent, streamlining the path to the solution.