Problem 33 - Algebra A Combined Function

Question

Hint: For Exercises 33 through 38 , first divide the equation through by the coefficient of \(x^{2}\) (or \(\left.y^{2}\right)\). $$ 3 x^{2}+3 y^{2}=75 $$

Step-by-Step Solution

Answer

The solution is a circle centered at (0,0) with radius 5.

Step 1: Identify the Coefficient of the Square Term

In the equation, identify the coefficient of the \(x^2\) and \(y^2\) terms. Both terms have the coefficient 3.

Step 2: Divide the Entire Equation by the Coefficient

To simplify, divide the entire equation by 3: \[ \frac{3x^2}{3} + \frac{3y^2}{3} = \frac{75}{3} \] This simplifies to: \[ x^2 + y^2 = 25 \].

Step 3: Recognize the Equation as a Circle

The equation now, \(x^2 + y^2 = 25\), is a standard form of the equation of a circle. This is centered at the origin \((0,0)\) with a radius \(r = \sqrt{25} = 5\).

Key Concepts

Equation Simplification

When you're dealing with equations, simplification is your friend. Simplifying an equation often means making it easier to understand and work with. In this exercise, the key to simplification was dividing every term by the coefficient of the squared terms.
Here's why: The equation given was quite complex due to the coefficient of 3 with each square term. By dividing every term by 3, you can transform the equation into a more manageable form. This step involves basic arithmetic operations that keep the equation balanced. Remember: whatever you do to one side of the equation, do the same to the other.
Simplifying helps in:

• Making the equation easier to interpret and solve.
• Allowing you to recognize patterns, like moving to the standard form.

Standard Form of Circle

The standard form of a circle's equation is a powerful way to express a circle in mathematics. Once we achieved simplification of our original equation, we reached the standard form: \[x^2 + y^2 = r^2\]
This form provides immediate insights:

• The center of the circle: For the above form, it is at the origin, \((0, 0)\).
• The radius of the circle: This is the square root of the number on the right-hand side. Here, that's \(\sqrt{25} = 5\).

Recognizing the standard form makes graphing and further calculations straightforward. It's a template that allows you to easily describe any circle.

Graphing Circles

Graphing a circle from its equation helps you visualize its size and position in the coordinate plane. Once you have the equation \(x^2 + y^2 = 25\), you're ready to graph.
Here's how you can approach it:

• **Center the circle**: Since the equation came down to the simple standard form, place the center at the origin \((0,0)\).
• **Determine the radius**: From the simplified equation, the radius is \(5\). This means every point on your circle is 5 units away from the center.
• **Draw the circle**: Use the radius to mark points at equal distances to form a round shape.

Following these steps, you can accurately sketch the circle, helping you to better understand the relationship between the algebraic equation and its geometric representation.