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Chapter 2: Problem 15

Use technology to obtain approximate solutions graphically. All solutions should be accurate to one decimal place. \(\begin{aligned} 2 x+8 y &=10 \\ x+y &=5 \end{aligned}\)

Short Answer

Expert verified
The approximate solution to the system of linear equations \(2x + 8y = 10\) and \(x + y = 5\) can be found graphically by plotting the two lines and finding their intersection point. After plotting, the two lines intersect at a point with coordinates approximately equal to \((2.0, 3.0)\). Therefore, the approximate solution is \(x \approx 2.0\) and \(y \approx 3.0\).

Step by step solution

01

Plot both equations on a Cartesian Plane

Plot the two equations, \(2x + 8y = 10\) and \(x + y = 5\), on a Cartesian plane. To do this, first express y in terms of x for the two equations. For the first equation, \(2x + 8y = 10 \Rightarrow y = \frac{10 - 2x}{8} = \frac{5 - x}{4}\) For the second equation, \(x + y = 5 \Rightarrow y = 5 - x\) Draw the two lines corresponding to these equations on the Cartesian plane.
02

Find the intersection point

Observe where the two lines intersect. This intersection point is the solution of the given system of linear equations. If necessary, you can use graphing software or a graphing calculator to plot the two lines and find the intersection.
03

Determine the solution

After graphing, you will see that the two lines intersect at a point with coordinates approximately equal to \((2.0, 3.0)\). So, the approximate solution to the system of linear equations is: \(x \approx 2.0\) \(y \approx 3.0\) Therefore, the approximate solution of the given system of linear equations is \((2.0, 3.0)\).

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Key Concepts

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

System of Linear Equations
A system of linear equations consists of two or more equations with multiple variables. In this case, we focus on visualizing two equations:
  • \(2x + 8y = 10\)
  • \(x + y = 5\)
The goal is to find the values of the variables that satisfy both equations simultaneously. These solutions are also known as the intersection points of the two lines represented by these equations in a graph.
Graphically, this means plotting each equation on a Cartesian plane and determining the point(s) where they intersect. To do this effectively, we first rewrite each equation to express \(y\) in terms of \(x\), enabling easy plotting of the lines.
Intersection Point
The intersection point is key to solving a system of linear equations graphically. It represents the values of \(x\) and \(y\) that satisfy both equations simultaneously.
When plotting
  • \(y = \frac{5 - x}{4}\)
  • \(y = 5 - x\)
we look for the point where these two lines cross on the Cartesian plane. This intersection point gives us the solution to the equations.
For accurate graphing, you can use graphing calculators or software, which help visualize and find the exact coordinates of the intersection. In this problem, this intersection point approximately appears at \((2.0, 3.0)\).
This point represents the solution: both equations are true when \(x\) is approximately 2.0 and \(y\) is approximately 3.0.
Approximate Solutions
In many real-world scenarios, obtaining an exact solution is not always possible. Graphing offers an excellent method to estimate solutions by providing a clear visual depiction.
Through graphing, you estimate where the solution lies. For the provided equations, the graphical method suggested that the lines crossed around the point \((2.0, 3.0)\). This solution is rounded to one decimal place, fitting the problem's requirement for an approximate solution.
For more precise solutions, especially when coordinates are not exact integers or are intricate, graphing software can enhance the accuracy of this approximation by zooming into the intersection point.
  • Utilize graphing technology to close in on the intersection
  • Ensure calculations align with the observed graphical solution
Using graphical methods offers insightful estimations, making them invaluable when looking for solutions that are adequate under constraints of time and tools.

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Most popular questions from this chapter

In 2007, total revenues from sales of recorded rock, religious, and classical music amounted to \(\$ 4\) billion. Religious music brought in twice as much as classical music, and rock music brought in \(\$ 3\) billion more than religious music. \({ }^{12}\) How much revenue was earned in each of the three categories of recorded music?

Use Gauss-Jordan row reduction to solve the given systems of equation. We suggest doing some by hand, and others using technology. HINT [See Examples 1-6.] $$ \begin{aligned} x-\frac{1}{2} y &=0 \\ \frac{1}{2} x-\quad \frac{1}{2} z &=-1 \\ 3 x-y-z &=-2 \end{aligned} $$

Big Red Bookstore wants to ship books from its warehouses in Brooklyn and Queens to its stores, one on Long Island and one in Manhattan. Its warehouse in Brooklyn has 1,000 books and its warehouse in Queens has 2,000 . Each store orders 1,500 books. It costs \(\$ 1\) to ship each book from Brooklyn to Manhattan and \(\$ 2\) to ship each book from Queens to Manhattan. It costs \(\$ 5\) to ship each book from Brooklyn to Long Island and \(\$ 4\) to ship each book from Queens to Long Island. a. If Big Red has a transportation budget of \(\$ 9,000\) and is willing to spend all of it, how many books should Big Red ship from each warehouse to each store in order to fill all the orders? b. Is there a way of doing this for less money? HINT [See Example 4.]

A bagel store orders cream cheese from three suppliers, Cheesy Cream Corp. (CCC), Super Smooth \& Sons (SSS), and Bagel's Best Friend Co. (BBF). One month, the total order of cheese came to 100 tons (they do a booming trade). The costs were \(\$ 80, \$ 50\), and \(\$ 65\) per ton from the three suppliers respectively, with total cost amounting to \(\$ 5,990\). Given that the store ordered the same amount from \(\mathrm{CCC}\) and \(\mathrm{BBF}\), how many tons of cream cheese were ordered from each supplier?

Law Enormous State University's campus publication, The Campus Inquirer, ran a total of 10 expos茅s five years ago, dealing with alleged recruiting violations by the football team and with theft by the student treasurer of the film society. Each expos茅 dealing with recruiting violations resulted in a \(\$ 4\) million libel suit, and the treasurer of the film society sued the paper for \(\$ 3\) million as a result of each expos茅 concerning his alleged theft. Unfortunately for The Campus Inquirer, all the lawsuits were successful, and the paper wound up being ordered to pay \(\$ 37\) million in damages. (It closed down shortly thereafter.) How many of each type of expos茅 did the paper run?
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