Question

find the solution set for the following system by graphing both of the system’s equations in the same rectangular coordinate system and finding all points of intersection. check all solutions in both equations.
\\(\

$$\begin{cases} x^2 + y^2 = 1 \\\\ x^2 + 16y^2 = 16 \\end{cases}$$

\\)

graph both of the system’s equations in the same rectangular coordinate system. choose the correct graph below.
options a, b, c, d with graphs

select the correct choice below and fill in any answer box to complete your choice.
\\(\bigcirc\\) a. the solution set is \\(\boxed{}\\)
(simplify your answer. type an ordered pair. use a comma to separate answers as needed.)
\\(\bigcirc\\) b. there is no solution.

Explanation:

Step1: Analyze the first equation

The equation \(x^{2}+y^{2} = 1\) represents a circle with center \((0,0)\) and radius \(r = 1\) (since the standard form of a circle is \((x - h)^{2}+(y - k)^{2}=r^{2}\), here \(h = 0,k = 0,r = 1\)).

Step2: Analyze the second equation

The equation \(x^{2}+16y^{2}=16\) can be rewritten by dividing both sides by 16: \(\frac{x^{2}}{16}+\frac{y^{2}}{1}=1\). This represents an ellipse with center \((0,0)\), semi - major axis \(a = 4\) (along the \(x\) - axis, since the denominator under \(x^{2}\) is larger) and semi - minor axis \(b = 1\) (along the \(y\) - axis).

Step3: Find the intersection points

We can solve the system of equations \(

$$\begin{cases}x^{2}+y^{2}=1\\x^{2}+16y^{2}=16\end{cases}$$

\) by subtracting the first equation from the second equation.
\((x^{2}+16y^{2})-(x^{2}+y^{2})=16 - 1\)
\(x^{2}+16y^{2}-x^{2}-y^{2}=15\)
\(15y^{2}=15\)
\(y^{2}=1\), so \(y=\pm1\)

When \(y = 1\), substitute into the first equation \(x^{2}+1^{2}=1\), then \(x^{2}=0\), so \(x = 0\).

When \(y=- 1\), substitute into the first equation \(x^{2}+(-1)^{2}=1\), then \(x^{2}=0\), so \(x = 0\).

So the intersection points are \((0,1)\) and \((0, - 1)\).

Now, looking at the graphs:

• The circle \(x^{2}+y^{2}=1\) has a small radius (radius 1) and the ellipse \(\frac{x^{2}}{16}+y^{2}=1\) is a horizontal ellipse with semi - major axis 4. The intersection points are on the \(y\) - axis (\(x = 0\)) at \((0,1)\) and \((0,-1)\). We need to find the graph that shows a circle of radius 1 and an ellipse with semi - major axis 4 intersecting at \((0,1)\) and \((0, - 1)\).

Assuming the graphs are labeled such that the correct graph shows the circle and the ellipse intersecting at the top and bottom of the circle (on the \(y\) - axis). Let's assume the correct graph is the one where the circle and the ellipse intersect at \((0,1)\) and \((0, - 1)\).

For the solution set, from our calculation, the solution set is \(\{(0,1),(0, - 1)\}\)

Answer:

The solution set is \(\{(0, 1),(0, - 1)\}\) (and the correct graph should show a circle of radius 1 and an ellipse with semi - major axis 4 intersecting at \((0,1)\) and \((0, - 1)\))