Find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations.
$\begin{cases}\) $\frac{x^2}{25}$ + $\frac{y^2}{9}$ = 1 \\ y = 3 \(\end{cases}$

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Identify the two equations in the system: the first is an ellipse given by $\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1$, and the second is a horizontal line given by $y = 3$.

Graph the ellipse by recognizing its center at the origin $(0,0)$, with a horizontal radius (semi-major axis) of 5 and a vertical radius (semi-minor axis) of 3, since $\sqrt{25} = 5$ and $\sqrt{9} = 3$.

Graph the line $y = 3$, which is a horizontal line crossing the y-axis at 3.

Find the points of intersection by substituting $y = 3$ into the ellipse equation: replace $y$ with 3 in $\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1$ to get $\frac{x^{2}}{25} + \frac{3^{2}}{9} = 1$.

Solve the resulting equation for $x^{2}$ to find the $x$-coordinates of the intersection points, then write the solution set as the points $(x, 3)$. Finally, check these points by substituting back into both original equations to verify they satisfy the system.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Graphing Conic Sections (Ellipses)

An ellipse is a conic section defined by an equation of the form (x²/a²) + (y²/b²) = 1, where a and b are the lengths of the semi-major and semi-minor axes. Graphing an ellipse involves plotting points that satisfy this equation, showing its shape centered at the origin or a shifted center. Understanding the ellipse's dimensions helps visualize where it intersects other graphs.

Graphing Linear Equations

A linear equation like y = 3 represents a horizontal line crossing the y-axis at 3. Graphing this line involves drawing a straight line parallel to the x-axis at the given y-value. Recognizing this helps in identifying intersection points with other curves or lines on the coordinate plane.

Finding Points of Intersection

Points of intersection occur where two graphs share the same coordinates, satisfying both equations simultaneously. To find these points, substitute the linear equation into the ellipse equation and solve for x or y. Checking these solutions in both equations confirms their validity as solutions to the system.