Ch. 7 - Conic Sections

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 7 - Conic Sections

Problem 63

Chapter 8, Problem 63

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.
${\begin{matrix} {(y - 2)}^{2} = x + 4 \\ y = - (\frac{1}{2}) x \end{matrix}$

Verified step by step guidance

Rewrite the first equation

(^{y - 2)} 2 = x + 4

to express

x

in terms of

y

. Subtract 4 from both sides to get

x = (^{y - 2)} 2 - 4

The second equation is already solved for

y

y = -rac{1}{2} x

. This is a linear equation representing a straight line.

Graph the parabola from step 1 by plotting points for various values of

y

and calculating corresponding

x

values using

x = (y - 2)^2 - 4

. This will give you the shape of the parabola on the coordinate plane.

Graph the line

y = -rac{1}{2} x

by choosing values for

x

and finding corresponding

y

values. Plot these points and draw the line.

Identify the points where the parabola and the line intersect on the graph. These intersection points are the solutions to the system. Substitute these points back into both original equations to verify they satisfy both equations.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Graphing Equations

Graphing involves plotting points that satisfy an equation on the coordinate plane. For this system, one equation is nonlinear (a parabola) and the other is linear. Understanding how to graph both accurately helps visualize their intersection points, which represent the solutions.

Solving Systems of Equations by Graphing

A system's solution set consists of points that satisfy all equations simultaneously. Graphing both equations on the same axes allows identification of intersection points, which correspond to these solutions. This method provides a visual approach to solving systems.

Checking Solutions in Both Equations

After finding intersection points, substituting them back into both original equations verifies their validity. This step ensures that the solutions satisfy both equations, confirming the accuracy of the graphing method and ruling out extraneous points.

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Related Practice

Textbook Question

In Exercises 57–62, use the vertex and the direction in which the parabola opens to determine the relation's domain and range. Is the relation a function?

$x = - 4 (y - 1)^{2} + 3$

Textbook Question

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}$

Textbook Question

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}\)x^2 + y^2 = 1 $\x$^2 + 9y^2 = 9\(\end{cases}$

Textbook Question

In Exercises 57–62, use the vertex and the direction in which the parabola opens to determine the relation's domain and range. Is the relation a function?

$y = - x^{2} + 4 x - 3$

Textbook Question

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.

${\begin{matrix} x = y^{2} - 3 \\ x = y^{2} - 3 y \end{matrix}$

Textbook Question

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}\)4x^2 + y^2 = 4 \\2x - y = 2\(\end{cases}$