Final answer:
To find two explicit functions by solving the equation for y in terms of x, divide both sides of the equation by 576 to simplify it. Then, isolate the y variable by solving for y in terms of x.
Step-by-step explanation:
To find two explicit functions by solving the equation for y in terms of x, we start with the equation 9x² + 64y² = 576.
First, divide both sides of the equation by 576 to simplify it: x²/64 + y²/9 = 1.
Now, solve for y in terms of x by isolating the y variable: y = ±(9/8)√(64 - 64x²).
To obtain explicit functions from the equation (9x² + 64y² = 576), a series of algebraic manipulations are undertaken. Initially, the equation is simplified by dividing both sides by 576, resulting in (x²/64 + y²/9 = 1). Subsequently, solving for (y) involves isolating the (y) variable. Taking the square root of both sides yields two solutions: (y = ± 3/4sqrt{64 - 64x²}. This implies two explicit functions for (y) in terms of (x), reflecting the positive and negative roots.
The equation represents an ellipse centered at the origin with major and minor axes aligned with the coordinate axes. The derived functions encapsulate the vertical displacement of points on the ellipse relative to the x-axis. Consequently, the manipulation process unveils the mathematical representation of the ellipse in terms of (x) and (y).