/*! 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 52 Use a graphing utility to determ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 52

Use a graphing utility to determine whether the system of equations has one solution, two solutions, or no solution. $$\left\\{\begin{aligned}-\frac{1}{2} x+y &=-1 \\ 7 x+y &=2 \end{aligned}\right.$$

Short Answer

Expert verified
The number of solutions will be determined by the interaction of the two plotted lines on the graph. It could either be one solution (if the lines intersect), two solutions (if the lines overlap), or no solution (if the lines are parallel). The exact answer will depend on the graph produced by the specific graphing utility used.

Step by step solution

01

Formulate the Equations for Graphing

Rewrite the equations in the form \(y = mx + c\), where \(m\) represents the gradient, \(c\) the y-intercept, \(x\) the x-coordinate, and \(y\) the y-coordinate. The first equation becomes \(y = 0.5x + 1\). The second equation is \(y = -7x + 2\).
02

Plot the Equations

Using a graphing utility (like Desmos, GeoGebra, or even a graphing calculator), plot both equations on the same graph. It's standard practice to use different colors for each equation to avoid confusion.
03

Analyze the Graph

Check how the graphs of the two equations interact. There are three scenarios to consider: the lines intersecting at a single point (one solution), the lines overlapping each other (infinite number of solutions or the system being dependent), or the lines are parallel having no intersections (no solution).

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.

Understanding Graphing Utilities
Graphing utilities are tools that help you visualize mathematical equations and functions. They are especially useful for solving systems of equations by graphing. When you graph equations, these utilities allow you to see how different lines interact. This is particularly helpful in determining the number of solutions a system of linear equations has. Some popular graphing utilities include:
  • Desmos: An interactive online graphing calculator that's user-friendly and great for beginners.
  • GeoGebra: A more advanced graphing tool that provides additional features and is useful for more complex equations.
  • Graphing calculators: Devices that can plot graphs, solve equations, and perform various mathematical operations.
When using these tools, it's a smart approach to plot each equation in a different color. This makes it easier to track and analyze how the graphs overlap or intersect.
Exploring Linear Equations
Linear equations are equations of the first degree, which means they graph as straight lines. They have the general form of \( y = mx + c \), where:
  • \( m \) is the slope or gradient of the line, indicating its steepness and direction.
  • \( c \) is the y-intercept, which is the point where the line crosses the y-axis.
In our exercise, the linear equations involved are \( y = 0.5x + 1 \) and \( y = -7x + 2 \). The slope \( m \) tells us:
  • If \( m > 0 \), the line is rising.
  • If \( m < 0 \), the line is falling.
  • If \( m = 0 \), the line is horizontal.
Understanding these components helps you quickly determine the shape and tilt of each line when graphing.
Solution Analysis for Systems of Equations
Analyzing the graph of a system of equations allows us to understand how many solutions exist. Depending on the interaction of the graphed lines, there are typically three possible scenarios:
  • **One Solution**: The lines intersect at exactly one point. This point is the solution to the system of equations, where both equations are satisfied simultaneously.
  • **No Solution**: The lines are parallel and do not intersect. This means there is no set of \( x \) and \( y \) values that satisfy both equations, and the system is inconsistent.
  • **Infinite Solutions**: The lines overlap completely, meaning they are the same line. Every point on the line is a solution, and the system is dependent.
In our exercise, using a graphing utility revealed that the lines intersect at a single point, indicating that the system has one unique solution. This visual confirmation is both effective and straightforward, making graphing a practical method for solving systems of equations.

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

You have a total of $$\$ 500,000$$ that is to be invested in (1) certificates of deposit, (2) municipal bonds, (3) blue-chip stocks, and (4) growth or speculative stocks. How much should be put in each type of investment? The certificates of deposit pay \(3 \%\) simple annual interest, and the municipal bonds pay \(10 \%\) simple annual interest. Over a five-year period, you expect the blue-chip stocks to return \(12 \%\) simple annual interest and the growth stocks to return \(15 \%\) simple annual interest. You want a combined annual return of \(10 \%\) and you also want to have only one-fourth of the portfolio invested in stocks.

Optimal Cost A humanitarian agency can use two models of vehicles for a refugee rescue mission. Each model A vehicle costs $$\$ 1000$$ and each model B vehicle costs $$\$ 1500$$. Mission strategies and objectives indicate the following constraints. \- A total of at least 20 vehicles must be used. 4\. A model A vehicle can hold 45 boxes of supplies. A model B vehicle can hold 30 boxes of supplies. The agency must deliver at least 690 boxes of supplies to the refugee camp. \- A model A vehicle can hold 20 refugees. A model \(\mathrm{B}\) vehicle can hold 32 refugees. The agency must rescue at least 520 refugees. What is the optimal number of vehicles of each model that should be used? What is the optimal cost?

Reasoning When solving a linear programming problem, you find that the objective function has a maximum value at more than one vertex. Can you assume that there are an infinite number of points that will produce the maximum value? Explain your reasoning.

Graphical Reasoning Two concentric circles have radii \(x\) and \(y\), where \(y>x .\) The area between the circles must be at least 10 square units. (a) Find a system of inequalities describing the constraints on the circles. (b) Use a graphing utility to graph the system of inequalities in part (a). Graph the line \(y=x\) in the same viewing window. (c) Identify the graph of the line in relation to the boundary of the inequality. Explain its meaning in the context of the problem.

The given linear programming problem has an unusual characteristic. Sketch a graph of the solution region for the problem and describe the unusual characteristic. Find the maximum value of the objective function and where it occurs. Objective function: \(z=x+y\) Constraints: \(x \geq 0\) \(y \geq 0\) \(-x+y \leq 1\) \(-x+2 y \leq 4\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

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.