Problem 3 - Precalculus

Question

Solve the given system of nonlinear equations. Sketch the graph of both equations on the same set of axes to verify the solution set. $$ \left\\{\begin{aligned} x^{2}+y^{2} &=16 \\ 16 x^{2}+4 y^{2} &=64 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The solutions are \((0, 4)\) and \((0, -4)\).

Step 1: Analyze Each Equation

The first equation, \(x^2 + y^2 = 16\), is a circle centered at the origin with a radius of 4. The second equation, \(16x^2 + 4y^2 = 64\), can be simplified to \(4x^2 + y^2 = 16\), which represents an ellipse centered at the origin with semi-major axis 4 along the x-axis and semi-minor axis 2 along the y-axis.

Step 2: Simplify the System of Equations

To solve the system, we can express both equations equivalently. The circle is already in standard form. For the ellipse, divide every term by 16 to get \(x^2/4 + y^2/16 = 1\). This makes it easier to handle the interaction between the shapes.

Step 3: Substitute and Solve

Substitute the circle equation \(x^2 + y^2 = 16\) into the modified ellipse equation. This can be simplified further: \(x^2/4 + (16 - x^2)/16 = 1\). Simplify the equation: \(x^2/4 + 1 - x^2/16 = 1\). Simplify to find the values of \(x\).

Step 4: Solve for Intersection Points

Continue simplifying to find that \(4x^2 - x^2 = 0\), leading to \(3x^2 = 0\), thus \(x = 0\). Next, substitute \(x = 0\) back into the circle equation to find \(y\)-values: \(y^2 = 16\), so \(y = 4\) or \(y = -4\).

Step 5: Verify Solution by Graphing

Both graphs intersect at points where \(x = 0\), \(y = 4\), and \(y = -4\). Sketch the circle and the ellipse on the same coordinate plane to see these intersections at \((0, 4)\) and \((0, -4)\), confirming these are the only points of intersection.

Key Concepts

intersection points

Intersection points are the solutions to the system of equations where the graphs meet on the coordinate plane. In this exercise, we have a circle and an ellipse. By solving the equations

• \( x^2 + y^2 = 16 \)
• \( 4x^2 + y^2 = 16 \)

we find that the solutions, or intersection points, have some common coordinates.
After solving, we determined that the intersection points are at \((0, 4)\) and \((0, -4)\). This means that these points lie on both the circle and the ellipse simultaneously.
It's essential to verify these points because they represent the precise locations where both figures overlap on the coordinate plane. Graphing helps us visually confirm these solutions, showing the points where the two shapes intersect.

graphing circles and ellipses

Graphing circles and ellipses requires understanding their standard equations. A circle's equation is often in the form \(x^2 + y^2 = r^2\), where \(r\) is the radius. Here, the circle's radius is 4, centered at the origin (0, 0).
For an ellipse, the standard form is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Simplifying the ellipse equation in this problem gives \(\frac{x^2}{4} + \frac{y^2}{16} = 1\), indicating that the semi-major axis is 4 and the semi-minor axis is 2.
While graphing:

• Locate the center of the circle and draw the radius outwards in all directions.
• For the ellipse, extend the semi-major and semi-minor axes from the center, respecting their lengths.

These guidelines help create an accurate visual representation aiding in identifying intersection points.

solving nonlinear equations

Solving nonlinear equations involves finding values that satisfy all equations in the system. A crucial step is expressing equations in a comparable form. For this non-linear system:

• Circle: \(x^2 + y^2 = 16\)
• Ellipse: \(4x^2 + y^2 = 16\)

Notice the simplification and transformation steps, such as dividing terms to place the ellipse in the form \(\frac{x^2}{4} + \frac{y^2}{16} = 1\) for easier manipulation.
Then, substitute one equation into another to eliminate variables, solving step by step for \(x\) and \(y\). Careful algebraic manipulation is key.
When you solve for every variable, these solutions should satisfy all original equations.

coordinate plane sketching

Coordinate plane sketching is an effective way to visualize solutions, providing a physical impression of mathematical concepts. Begin by drawing both the x-axis and y-axis, each evenly marked.
Plot each graph one at a time:

• Start with the circle by marking the center, drawing the radius out equally in all directions.
• Next, move to the ellipse, sketching using the semi-major and semi-minor axes as guides.

The sketching verifies the algebraic solutions, pinpointing intersection coordinates. Ensure axes are balanced to maintain scale.
Accurate sketching ensures you see where the graphs intersect, thus confirming solutions like \((0, 4)\) and \((0, -4)\).