Problem 16 - Intermediate Algebra

Question

Write each denominator in the equation \(\frac{x^{2}}{81}+\frac{y^{2}}{49}=1\) as the square of a number.

Step-by-Step Solution

Answer

Rewrite the equation as \(\frac{x^{2}}{9^2}+\frac{y^{2}}{7^2}=1\).

Step 1: Identify the Denominators

The given equation is \(\frac{x^{2}}{81}+\frac{y^{2}}{49}=1\). Here, the denominators are 81 and 49.

Step 2: Express 81 as a Square

To express 81 as a square, note that \(81 = 9 \times 9\), or \(81 = 9^2\).

Step 3: Express 49 as a Square

Similarly, express 49 as a square: \(49 = 7 \times 7\), or \(49 = 7^2\).

Step 4: Rewrite the Equation

Substitute the squared values into the equation: \(\frac{x^{2}}{9^2}+\frac{y^{2}}{7^2}=1\).

Key Concepts

Denominator Identification

Knowing how to identify the denominators in an equation is a fundamental skill, especially when dealing with equations of shapes like ellipses. In our example equation, \(\frac{x^{2}}{81}+\frac{y^{2}}{49}=1\), identifying the denominators means recognizing the values 81 and 49. These numbers, found underneath the fractions, act as divisors for the squared terms \(x^2\) and \(y^2\).

This step is crucial because it sets the pathway for further transformations. Recognizing these denominators allows you to determine the scale of each axis of the ellipse when the equation is in a standard form. Always remember:

• Denominators are numbers that the variables are being divided by.
• They determine how each variable is scaled.

Spotting these denominators early in the problem-solving process will make subsequent steps significantly easier and clearer.

Expressing Numbers as Squares

Transforming denominators into perfect squares can help simplify and transform an equation into a standard form, which is often easier to work with. In our example, we express the denominators of 81 and 49 as squares. Begin with 81, which can be expressed as \(9^2\) because \(9 \times 9 = 81\).

Similarly, express 49 as \(7^2\) since \(7 \times 7 = 49\).

Converting numbers into their square form is helpful because:

• It reveals the lengths of the semi-axes of an ellipse when the equation is transformed into its standard form.
• Makes the equation easier to understand when identifying geometric properties.

This technique is not only limited to solving ellipse equations but is also broadly used in many algebraic transformations.

Rewriting Equations

Rewriting equations is an important step in algebra, as it allows us to transform complex expressions into simpler, standard forms. In our original equation \(\frac{x^{2}}{81}+\frac{y^{2}}{49}=1\), after expressing the denominators as squares, we rewrite it as \(\frac{x^{2}}{9^2}+\frac{y^{2}}{7^2}=1\).

This step involves substituting each denominator with its respective square value. This method simplifies the understanding of the equation's geometry, especially for ellipses, whose standard form is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

By rewriting, we easily identify:

• The semi-length of the major axis as the square root of the larger denominator.
• The semi-length of the minor axis as the square root of the smaller denominator.

This rewriting process is key in moving from a general expression to a geometric interpretation.