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Ch. 7 - Conic Sections
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 7 - Conic SectionsProblem 65
Chapter 8, Problem 65

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.
{4x2+y2=42x−y=2\(\begin{cases}\)4x^2 + y^2 = 4 \\2x - y = 2\(\end{cases}\)

Verified step by step guidance
1
Rewrite the first equation \(4x^{2} + y^{2} = 4\) to understand its graph. This is an equation of an ellipse. You can express it in standard form by dividing both sides by 4: \(\frac{4x^{2}}{4} + \frac{y^{2}}{4} = 1\), which simplifies to \(x^{2} + \frac{y^{2}}{4} = 1\).
Rewrite the second equation \(2x - y = 2\) in slope-intercept form to make graphing easier. Solve for \(y\): \(y = 2x - 2\).
Graph both equations on the same coordinate plane: the ellipse \(x^{2} + \frac{y^{2}}{4} = 1\) and the line \(y = 2x - 2\). The points where the line intersects the ellipse are the solutions to the system.
To find the exact points of intersection algebraically, substitute \(y = 2x - 2\) into the ellipse equation: \(x^{2} + \frac{(2x - 2)^{2}}{4} = 1\).
Simplify the resulting equation and solve for \(x\). Then, use the values of \(x\) to find corresponding \(y\) values using \(y = 2x - 2\). These \((x, y)\) pairs are the solutions to the system. Finally, check each solution by substituting back into both original equations to verify correctness.

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Key Concepts

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

Graphing Conic Sections

Graphing conic sections involves plotting curves defined by quadratic equations, such as ellipses, parabolas, and hyperbolas. In this problem, the equation 4x² + y² = 4 represents an ellipse. Understanding how to sketch this curve helps visualize where it might intersect with other graphs.
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Geometries from Conic Sections

Graphing Linear Equations

Linear equations like 2x - y = 2 represent straight lines on the coordinate plane. By rewriting the equation in slope-intercept form (y = mx + b), you can easily plot the line and analyze its relationship with other graphs, such as points of intersection with conic sections.
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Categorizing Linear Equations

Solving Systems of Equations by Graphing

Solving systems by graphing involves plotting each equation on the same coordinate plane and identifying their intersection points. These points represent solutions that satisfy both equations simultaneously. Checking these points in both equations confirms their validity.
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Solving Systems of Equations - Substitution
Related Practice
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.

{x225+y29=1y=3\(\begin{cases}\]\frac{x^2}{25}\) + \(\frac{y^2}{9}\) = 1 \(\y\) = 3\(\end{cases}\)

Textbook Question

Graph each semiellipse. y = -√16 - 4x²

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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. {x=(y+2)2−1(x−2)2+(y+2)2=1\(\begin{cases}\)x = (y + 2)^2 - 1 \\(x - 2)^2 + (y + 2)^2 = 1\(\end{cases}\)

Textbook Question

Find the standard form of the equation of an ellipse with vertices at (0, -6) and (0, 6), passing through (2, 4).

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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.

{x=y2−3x=y2−3y\(\left\)\{\(\begin{array}{l}\)x=y^2-3\\ x=y^2-3y\(\end{array}\]\right\).

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.

{(y−2)2 =x+4y=−(12)x\(\left\)\{\(\begin{array}{l}\]\left\)(y-2\(\right\))^2\(\text{ }\)=x+4\\ y=-\(\text{(}\[\frac\)12\(\text{)}\)x\(\end{array}\]\right\).