Ch. 7 - Conic Sections

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Blitzer 8th Edition

Ch. 7 - Conic Sections

문제 65

8장, 문제 65

In Exercises 63–68, 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.
${\begin{matrix} x = y^{2} - 3 \\ x = y^{2} - 3 y \end{matrix}$

검증된 단계별 안내

Rewrite both equations to clearly identify the relationships: the first equation is

x = y^{2} - 3

and the second equation is

x = y^{2} - 3 y

Since both equations express

x

in terms of

y

, set the right-hand sides equal to each other to find the values of

y

where the graphs intersect:

y^{2} - 3 = y^{2} - 3 y

Simplify the equation by subtracting

y^{2}

from both sides, resulting in

-3 = -3y

. Then solve for

y

by dividing both sides by -3.

Find the corresponding

x

values by substituting each

y

value back into either original equation, for example,

x = y^{2} - 3

Verify each solution pair

(x, y)

by substituting into both original equations to ensure they satisfy both, confirming the points of intersection and thus the solution set.

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주요 개념

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

Graphing Equations in the Coordinate Plane

Graphing involves plotting points that satisfy an equation on the rectangular coordinate system. For each equation, you find pairs (x, y) that make the equation true and plot them to visualize the curve or line. This helps in identifying where two graphs intersect.

Systems of Equations and Intersection Points

A system of equations consists of two or more equations with the same variables. The solution set is the set of points that satisfy all equations simultaneously, often found where their graphs intersect. Intersection points represent these common solutions.

Checking Solutions by Substitution

After finding potential solutions from the graph, substitute the coordinates back into each original equation to verify they satisfy both. This step ensures the accuracy of the solution set and confirms that the intersection points are valid.