Problem 150 - Intermediate Algebra

Question

In the following exercises, simplify. $$ -\frac{30 x^{2}}{105 y^{2}} $$

Step-by-Step Solution

Answer

-\frac{2 x^{2}}{7 y^{2}}

Step 1: Find the greatest common divisor (GCD)

Identify the greatest common divisor of the numerator and the denominator. In this case, the GCD of 30 and 105 is 15.

Step 2: Simplify the coefficients

Divide both the numerator and the denominator by the GCD found in Step 1.\[-\frac{30 x^{2}}{105 y^{2}} = -\frac{30 \div 15 \, x^{2}}{105 \div 15 \, y^{2}} = -\frac{2 x^{2}}{7 y^{2}}divide each term by 15\]

Step 3: Write the simplified fraction

Combine the simplified coefficients and the variables to express the fraction in its simplest form:\[-\frac{2 x^{2}}{7 y^{2}}\]

Key Concepts

Greatest Common Divisor (GCD)

To simplify algebraic expressions, finding the greatest common divisor (GCD) is often crucial. The GCD helps us reduce fractions by showing the largest number that can divide both the numerator and the denominator without leaving a remainder.
For example, consider the given fraction: \[:-\frac{30 x^{2}}{105 y^{2}}\text{\].} To find the GCD of 30 and 105:

• List the factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
• List the factors of 105: 1, 3, 5, 7, 15, 21, 35, 105

Look for the largest common factor. Here, it's 15.
Once you identify the GCD, proceed to simplify the expression by dividing both the numerator and the denominator by this number.

Simplifying Fractions

Simplifying fractions is an integral part of algebra. It involves reducing the fraction to its lowest terms so that it's easier to understand and handle.
After identifying the GCD, divide the numerator and the denominator by this number. Using our example: \[-\frac{30 x^{2}}{105 y^{2}}\text{ = }-\frac{30 \text{ \textbackslash div 15 } x^{2}}{105 \text{ \textbackslash div 15 } y^{2}}\text{ = }-\frac{2 x^{2}}{7 y^{2}}\]
The expression becomes \[-\frac{2 x^{2}}{7 y^{2}}\text{\], making it more manageable.
This method can be applied to any fraction involving algebraic terms.

Algebraic Simplification

Algebraic simplification involves reducing an expression to its simplest form. This can make problem-solving easier and results more interpretable.
Use the GCD to reduce coefficients and cancel common variables. Let’s break down the steps using the provided exercise. \[:-\frac{30 x^{2}}{105 y^{2}}\text{ \]
Step 1: Identify the GCD (in this case, 15).
Step 2: Divide both the numerator and the denominator by the GCD. Using algebraic terms we get: \[-\frac{30 \text{ \textbackslash div 15 } x^{2}}{105 \text{ \textbackslash div 15 } y^{2}}\]. This results in: \[-\frac{2 x^{2}}{7 y^{2}}.\]
Combining the numerals and variables in their simplest form gives us the final simplified expression.

• Always start with identifying and factoring out greatest common numerators.
• Lastly, combine the simplified terms to get the simplest form.

Mastering these techniques will help you navigate complex algebraic expressions with greater ease.