/*! 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 38 Exer \(37-40:\) Find the points ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 38

Exer \(37-40:\) Find the points of intersection of the graphs of the equations. Sketch both graphs on the same coordinate plane, and show the points of intersection. $$\left\\{\begin{array}{l} 5 x^{2}+y^{2}=189 \\ 3 x+y=7 \end{array}\right.$$

Short Answer

Expert verified
The points of intersection are (5, -8) and (-2, 13).

Step by step solution

01

Express One Variable in Terms of the Other

From the linear equation \(3x + y = 7\), express \(y\) in terms of \(x\):\[y = 7 - 3x\]
02

Substitute Linear Equation into Quadratic Equation

Substitute \(y = 7 - 3x\) into the quadratic equation \(5x^2 + y^2 = 189\):\[5x^2 + (7 - 3x)^2 = 189\]
03

Expand and Simplify the Equation

Expand the equation \((7 - 3x)^2\) and simplify:\[(7 - 3x)^2 = 49 - 42x + 9x^2\]Substitute back:\[5x^2 + 49 - 42x + 9x^2 = 189\]Combine like terms:\[14x^2 - 42x + 49 = 189\]
04

Move All Terms to One Side

Rearrange the equation and set it to zero:\[14x^2 - 42x + 49 - 189 = 0\]\[14x^2 - 42x - 140 = 0\]
05

Divide the Equation to Simplify

Divide the whole equation by 2 to simplify:\[7x^2 - 21x - 70 = 0\]
06

Use the Quadratic Formula

Apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) where \(a = 7\), \(b = -21\), and \(c = -70\):\[x = \frac{-(-21) \pm \sqrt{(-21)^2 - 4 \cdot 7 \cdot (-70)}}{2 \cdot 7}\]Calculate the discriminant \((-21)^2 + 4 \times 7 \times 70 = 441 + 1960 = 2401\).\[x = \frac{21 \pm \sqrt{2401}}{14}\]Calculate \(\sqrt{2401} = 49\).\[x = \frac{21 \pm 49}{14}\]
07

Find Values of x

Calculate each value of \(x\):\[x_1 = \frac{21 + 49}{14} = 5\]\[x_2 = \frac{21 - 49}{14} = -2\]
08

Substitute Back to Find y Values

Plug \(x_1 = 5\) and \(x_2 = -2\) back into the linear equation \(y = 7 - 3x\):For \(x_1 = 5\),\[y = 7 - 3 imes 5 = -8\]For \(x_2 = -2\),\[y = 7 - 3 \times (-2) = 13\]
09

Identify Points of Intersection

The points of intersection are \((5, -8)\) and \((-2, 13)\).
10

Sketch the Graphs

Draw the graph of the parabola \(5x^2 + y^2 = 189\), which is an ellipse, and the graph of the line \(3x+y=7\) on the same coordinate plane. The points \((5, -8)\) and \((-2, 13)\) are where they intersect.

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.

Quadratic Equation
A quadratic equation is a polynomial equation of degree two, which means it includes an
  • variable squared (e.g., \(x^2\))
  • can typically be written in the form \(ax^2 + bx + c = 0\)
In this exercise, the quadratic equation is part of the system of equations involving:
  • \(5x^2 + y^2 = 189\)
To solve a quadratic equation, one often uses various strategies, such as factoring, completing the square, or applying the quadratic formula. For more complex equations like this, substitution of known values can simplify the process. Breaking down the equation into simpler parts such as completing the square and making substitutions helps in finding the solutions to complex systems where quadratic equations are involved.
Linear Equation
Linear equations are equations of the first degree, which implies their graphs form straight lines.
  • The standard form of a linear equation in two variables is \(ax + by = c\)
  • In this exercise, it is \(3x + y = 7\)
Linear equations can be easily solved algebraically for one of the variables. You can discover this through rearrangement, enabling substitution into other equations, just like outlined in the given example with \(y = 7 - 3x\). Understanding how linear equations function within a system allows you to manipulate them effectively, facilitating the solution of the system by substitution or elimination. They are foundational in understanding more complex systems including both linear and non-linear equations.
Intersection Points
The intersection points of two graphs are the values of \(x\) and \(y\) that satisfy both equations simultaneously.
  • These points show where the two graphs meet on the graph
  • In this case, calculate them by solving the system of equations together
Through a systematic approach鈥攂y solving for \(x\) and substituting back to find \(y\)鈥攖he intersection points were discovered as
  • \((5, -8)\) and \((-2, 13)\)
Once calculated, these points reveal the solution to this real-world problem involving both a linear and a quadratic equation. Comprehending intersection points deeply is crucial, as they represent shared solutions in graphing systems of equations.
Graphing Systems of Equations
Graphing systems of equations provides a visual representation of where two or more equations meet or cross each other. It's a combination of graphical displays illustrating the:
  • Quadratic equation as an ellipse in this case
  • Linear equation as a straight line
On the coordinate plane, these intersections come to life at the calculated points of
  • \((5, -8)\) and \((-2, 13)\)
Sketching out these graphs aids in understanding complex algebraic solutions and brings clarity to the solutions derived from solving the equations algebraically. It鈥檚 an essential tool for comprehending multi-equation systems, enhancing the ability to visualize the mathematical concepts involved.

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

Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix. $$e=1, \quad r \cos \theta=5$$

Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix. $$e=\frac{1}{3}, \quad r=2 \sec \theta$$

Graph the polar equation for the indicated values of \(\theta,\) and use the graph to determine symmetries. $$r=2 \sin ^{2} \theta \tan ^{2} \theta ; \quad-\pi / 3 \leq \theta \leq \pi / 3$$

Show that $$x=a \cos t+h, \quad y=b \sin t+k ; \quad 0 \leq t \leq 2 \pi$$ are parametric equations of an ellipse with center \((h, k)\) and axes of lengths \(2 a\) and \(2 b\).

(a) Describe the graph of a curve \(C\) that has the parametrization $$x=3+2 \sin t, \quad y=-2+2 \cos t ; \quad 0 \leq t \leq 2 \pi$$ (b) Change the parametrization to $$x=3-2 \sin t, \quad y=-2+2 \cos t, \quad 0 \leq t \leq 2 \pi$$ and describe how this changes the graph from part (a). (c) Change the parametrization to $$x=3-2 \sin t, \quad y=-2-2 \cos t, \quad 0 \leq t \leq 2 \pi$$ and describe how this changes the graph from part (a).
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

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.