Problem 50 - Introductory Algebra for College Students

Question

Perform the indicated division or state that the expression is undefined. $$\frac{-60}{6}$$

Step-by-Step Solution

Answer

-10

Step 1: Perform Division

To solve this problem, divide -60 by 6. So, \(-60 \div 6 = -10\). Thus, the expression \(\frac{-60}{6}\) equals -10.

Step 2: Identify the algebraic structure

Determine the type of algebraic problem.

Step 3: Apply algebraic techniques

Use factoring, expanding, or systematic methods.

Step 4: Simplify and solve

Simplify expressions and solve for unknowns.

Step 5: State the result

Write the final answer.

Key Concepts

Simplifying Algebraic Expressions

When working with algebra, simplifying expressions is a fundamental skill. It involves reducing an expression to its simplest form, making it easier to understand and solve. To simplify an algebraic expression, combine like terms, which are terms that have the same variable raised to the same power. Perform operations with coefficients (the numerical parts) while keeping the variables unchanged.

For example, in the expression \(2x + 3x - 5x^2 + x^2\), to simplify, first combine like terms \(2x + 3x\) to get \(5x\), and then combine like terms \(x^2\) terms to get \( -4x^2 \). The simplified expression is \(5x - 4x^2\). This method is crucial as it lays the groundwork for further operations like the division of negative numbers.

Negative Numbers Division

Dividing negative numbers may seem tricky, but it follows a simple rule—when you divide a negative number by a positive number, the result is negative, and vice versa. This is because division is essentially repeated subtraction.

If you're dividing \( -60 \) by \( 6 \), think of it as subtracting \( 6 \) from \( 0 \) a total of \( -60 \) times. The answer is \( -10 \), meaning you had to subtract \( 6 \) ten times to reach \( -60 \). Since every subtraction diminishes the number, starting from \( 0 \), you end up in the negative number range, hence \( -10 \).

It is important to remember that the signs need to be considered during the division to ensure the correct result of an algebraic expression.

Undefined Expressions in Mathematics

In mathematics, an expression is undefined when it does not result in a real number or extends beyond the limits of the number system. A common example is division by zero. No number can be divided by zero, as this would imply that zero times some number equals the original number, which is impossible.

For example, the expression \( \frac{1}{0} \) is undefined because you cannot distribute \( 1 \) into zero parts. This concept is vital in understanding the limitations of algebraic operations and maintaining the integrity of mathematical laws.

However, an expression like \( \frac{-60}{6} \) is defined and can be simplified. In contrast, something like \( \frac{5}{0} \) would be an example of an undefined expression in mathematics.