Problem 3 - Elementary Algebra

Question

For the following problems, find \(-a\) if \(a\) is: $$ -\frac{8}{9} $$

Step-by-Step Solution

Answer

Answer: \(\frac{8}{9}\)

Step 1: Rewrite the problem

We are asked to find \(-a\), where \(a = -\frac{8}{9}\). So the problem can be rewritten as: $$ -(-\frac{8}{9}) $$

Step 2: Multiply by \(-1\)

Multiply the fraction \(-\frac{8}{9}\) by \(-1\): $$ -1 \times (-\frac{8}{9}) = \frac{(-1)(-8)}{9} $$

Step 3: Simplify the fraction

Now, simplify the fraction in the numerator: $$ \frac{(-1)(-8)}{9} = \frac{8}{9} $$ So, \(-a = \frac{8}{9}\).

Key Concepts

Algebraic Expressions

Algebraic expressions are like sentences in mathematics, made up of numbers, variables, and operations. In the exercise, the focus was on understanding the expression \( -a \). Here, \( a \) is given as \( -\frac{8}{9} \). The negative sign in front of \( a \) indicates that we need to find the opposite of \( a \).

• Variables such as \( a \) can represent any number in an expression.
• The negative sign changes the direction of the value, which is key when dealing with minus signs.
• Rewriting the problem can simplify the process of finding a solution. In this case, \( -(-\frac{8}{9}) \) becomes a simpler expression to work with.

By understanding how negative signs and variables interact, we can solve algebraic expressions efficiently. Rewriting them helps clarify their components and intended operations.

Fraction Simplification

Fraction simplification is a crucial skill that makes complex problems easier to handle. A simplified fraction uses the smallest possible numbers to convey the same value, making calculations more straightforward. This makes it easier to interpret the result of an operation, such as multiplication.

• In the solution, multiplying \( -\frac{8}{9} \) by \( -1 \) directly simplifies to \( \frac{8}{9} \).
• Negative signs in fractions are treated like any other component of the fraction and must follow rules of sign multiplication.
• When simplifying, ensure that the numerator and the denominator are both treated equally, following the rules of arithmetic operations.

Simplifying fractions helps in reducing errors and makes subsequent steps in calculations much easier to follow. This is particularly useful in algebra, where expressions can get complicated quickly.

Multiplying by Negative Numbers

Multiplying by negative numbers is an important mathematical concept, especially when working with algebraic expressions. The rule is simple: multiplying two numbers with the same sign results in a positive number, and multiplying two numbers with different signs results in a negative number.

• In our example, \( -1 imes -\frac{8}{9} = \frac{8}{9} \) shows how two negatives result in a positive.
• Understanding these rules is essential because they frequently occur in various types of mathematical problems including algebra.
• Maintaining the correct signs during calculations ensures accurate results and helps prevent simple errors that might affect complex problem-solving.

Whenever you encounter multiplication involving negative numbers, remember to carefully keep track of signs to maintain the integrity of your mathematical work.