/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 14 Graph each equation of the syste... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 14

Graph each equation of the system. Then solve the system to find the points of intersection. $$ \left\\{\begin{array}{r} x^{2}+y^{2}=8 \\ x^{2}+y^{2}+4 y=0 \end{array}\right. $$

Short Answer

Expert verified
The points of intersection are (2, -2) and (-2, -2).

Step by step solution

01

- Identify and rearrange the equations

The given system of equations is: 1) \( x^2 + y^2 = 8 \) 2) \( x^2 + y^2 + 4y = 0 \). Rewrite the second equation to isolate the terms involving \(y\): \( x^2 + y^2 + 4y = 0 \rightarrow x^2 + (y^2 + 4y) = 0 \).
02

- Complete the square for the second equation

Complete the square for the terms involving \(y\) in the second equation: \( y^2 + 4y \) can be written as \( (y+2)^2 - 4 \). Thus, the second equation becomes: \( x^2 + (y+2)^2 - 4 = 0 \rightarrow x^2 + (y+2)^2 = 4 \).
03

- Rewrite the equations in standard form

Now, rewrite both equations in standard form: 1) \( x^2 + y^2 = 8 \) 2) \( x^2 + (y+2)^2 = 4 \). These are equations of circles.
04

- Find the centers and radii of the circles

For the first circle, \( x^2 + y^2 = 8 \), the center is (0, 0) and the radius is \( \sqrt{8} = 2\sqrt{2} \). For the second circle, \( x^2 + (y+2)^2 = 4 \), the center is (0, -2) and the radius is \( \sqrt{4} = 2 \).
05

- Graph the circles

Graph both circles on the coordinate plane. The first circle with center (0,0) and radius \( 2\sqrt{2} \), and the second circle with center (0,-2) and radius 2. Identify the points where the circles intersect.
06

- Solve the system algebraically

Solve the system algebraically to find the exact points of intersection. Subtract the second equation from the first: \((x^2 + y^2) - (x^2 + (y+2)^2) = 8 - 4\) \( y^2 - (y+2)^2 = 4 \) \( y^2 - (y^2 + 4y + 4) = 4 \) \( -4y - 4 = 4 \) \( -4y = 8 \) \( y = -2 \). Substitute \( y = -2 \) back into the first equation: \( x^2 + (-2)^2 = 8 \) \( x^2 + 4 = 8 \) \( x^2 = 4 \) \( x = \pm 2 \). The points of intersection are \( (2, -2) \) and \( (-2, -2) \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Equations
When graphing equations, you plot their points on a coordinate plane. Every equation has a set of solutions that form a line or curve. In our system, we have two equations representing circles. The first equation is:
\( x^2 + y^2 = 8 \)
This represents a circle centered at the origin (0, 0) with a radius \( \text{radius} = \text{sqrt}(8) = 2\text{sqrt}(2) \).
To plot this circle:
  • Find the center point (0, 0).
  • Use your radius (approximately 2.83) and draw a circle around the center.
The second equation is:
\( x^2 + (y+2)^2 = 4 \)
This circle is centered at (0, -2) with a radius of 2.
To graph this circle:
  • Locate the center (0, -2).
  • Draw the circle with a radius of 2.
By graphing these circles, you will visually see the points where they intersect. These intersection points are the solutions to the system of equations.
Completing the Square
Completing the square is a method used to solve quadratic equations. It can also help to rewrite equations for graphing shapes like circles.
Let's take the second equation in our system:
\( x^2 + y^2 + 4y = 0 \)
We rearrange it to isolate the y terms:
\( x^2 + (y^2 + 4y) = 0 \)
To complete the square for \( y^2 + 4y \), we add and subtract 4:
\[ y^2 + 4y = (y+2)^2 - 4 \]
This makes our equation:
\[ x^2 + (y+2)^2 - 4 = 0 \]
Simplifying, we get:
\[ x^2 + (y+2)^2 = 4 \]
Now it's clear that this represents a circle centered at (0, -2) with a radius of 2. Completing the square helps us recognize and graph geometric shapes more easily.
Circles in Coordinate Geometry
In coordinate geometry, a circle's equation can be written in standard form:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Here, (h, k) is the center, and r is the radius.
For our first circle:
\[ x^2 + y^2 = 8 \]
The center is (0, 0), and the radius is \( \text{sqrt}(8) = 2\text{sqrt}(2) \).
For the second circle:
\[ x^2 + (y+2)^2 = 4 \]
The center is (0, -2), and the radius is 2.
Graphing both circles on the same coordinate plane helps us find their intersection points.
In circles, the tricky part isn't recognizing the shape but finding those points accurately using both graphing and algebraic methods.
Algebraic Solution of Systems
To find the points where two circles intersect, we use algebra. We have:
1) \( x^2 + y^2 = 8 \)
2) \( x^2 + (y+2)^2 = 4 \)
We subtract the second equation from the first:
  • n) \ [ x^2 + y^2 ] - [ x^2 + (y+2)^2 ] = 8 - 4 \America ) \Opponent [- (y^2 + 4y + 4) y^2 = 4\binz]
menyy^2 - _round] = -4y - 4
robin4y \ Americaiveline /> y = -2 )
Los \tex{2()}in (}) orxy ] = 4neging ewline{2 = 4)}water, x = two }

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Nutrition A dietitian wishes a patient to have a meal that has 66 grams (g) of protein, 94.5 g of carbohydrates, and 910 milligrams (mg) of calcium. The hospital food service tells the dietitian that the dinner for today is chicken, corn, and \(2 \%\) milk. Each serving of chicken has \(30 \mathrm{~g}\) of protein, \(35 \mathrm{~g}\) of carbohydrates, and \(200 \mathrm{mg}\) of calcium. Each serving of corn has \(3 \mathrm{~g}\) of protein, \(16 \mathrm{~g}\) of carbohydrates, and \(10 \mathrm{mg}\) of calcium. Each glass of \(2 \%\) milk has \(9 \mathrm{~g}\) of protein, \(13 \mathrm{~g}\) of carbohydrates, and \(300 \mathrm{mg}\) of calcium. How many servings of each food should the dietitian provide for the patient?

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{array}{rr} 2 x+y-3 z= & 0 \\ -2 x+2 y+z= & -7 \\ 3 x-4 y-3 z= & 7 \end{array}\right. $$

A young couple has \(\$ 25,000\) to invest. As their financial consultant, you recommend that they invest some money in Treasury bills that yield \(7 \%,\) some money in corporate bonds that yield \(9 \%,\) and some money in junk bonds that yield \(11 \% .\) Prepare a table showing the various ways that this couple can achieve the following goals: (a) \(\$ 1500\) per year in income (b) \(\$ 2000\) per year in income (c) \(\$ 2500\) per year in income (d) What advice would you give this couple regarding the income that they require and the choices available?

Presale Order A wireless store owner takes presale orders for a new smartphone and tablet. He gets 340 preorders for the smartphone and 250 preorders for the tablet. The combined value of the preorders is \(\$ 486,000 .\) If the price of a smartphone and tablet together is \(\$ 1665,\) how much does each device cost?

A Florida juice company completes the preparation of its products by sterilizing, filling, and labeling bottles. Each case of orange juice requires 9 minutes (min) for sterilizing, 6 min for filling, and 1 min for labeling. Each case of grapefruit juice requires 10 min for sterilizing, 4 min for filling, and 2 min for labeling. Each case of tomato juice requires 12 min for sterilizing, 4 min for filling, and 1 min for labeling. If the company runs the sterilizing machine for 398 min, the filling machine for 164 min, and the labeling machine for 58 min, how many cases of each type of juice are prepared?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.