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Chapter 8: Problem 72

will help you prepare for the material covered in the next section. Graph \(x-y=3\) and \((x-2)^{2}+(y+3)^{2}=4\) in the same rectangular coordinate system. What are the two intersection points? Show that each of these ordered pairs satisfies both equations.

Short Answer

Expert verified
The two intersection points can be found graphically. To verify, substitute these points into both original equations to ensure that they satisfy both.

Step by step solution

01

Plot the first equation

The first equation \(x-y=3\) is a linear equation. This can be rewritten as \(y=x-3\), indicating that the y-intercept is -3 and the slope is 1. This line can be drawn on the coordinate system accordingly.
02

Plot the second equation

The second equation \((x-2)^{2}+(y+3)^{2}=4\) is a circle. The center of the circle is at the point (2,-3) and the radius is 2 (since 2^2 = 4). Draw this circle on the same coordinate system.
03

Identify the intersection points

The two graphs intersect at two points. These can be found visually by following where the line and the circle intersect.
04

Verify the intersection points

The two intersection points should be substituted back into both original equations in order to verify that they satisfy both equations.

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Most popular questions from this chapter

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function Constraints $$ \begin{aligned} &z=2 x+3 y\\\ &\left\\{\begin{array}{l} {x \geq 0, y \geq 0} \\ {2 x+y \leq 8} \\ {2 x+3 y \leq 12} \end{array}\right. \end{aligned} $$

What is an objective function in a linear programming problem?

If \(x=3, y=2,\) and \(z=-3,\) does the ordered triple \((x, y, z)\) satisfy the equation \(2 x-y+4 z=-8 ?\)

Use the two steps for solving a linear programming problem, given in the box on page \(888,\) to solve the problems. On June \(24,1948,\) the former Soviet Union blocked all land and water routes through East Germany to Berlin. A gigantic airlift was organized using American and British planes to bring food, clothing, and other supplies to the more than 2 million people in West Berlin. The cargo capacity was \(30,000\) cubic feet for an American plane and \(20,000\) cubic feet for a British plane. To break the Soviet blockade, the Western Allies had to maximize cargo capacity but were subject to the following restrictions: \(\cdot\) No more than 44 planes could be used. \(\cdot\) The larger American planes required 16 personnel per flight, double that of the requirement for the British planes. The total number of personnel available could not exceed 512 . \(\cdot\) The cost of an American flight was 9000 and the cost of a British flight was 5000 . Total weekly costs could not exceed $300,000 Find the number of American and British planes that were used to maximize cargo capacity.

Will help you prepare for the material covered in the next section. In each exercise, graph the linear function. $$ f(x)=-2 $$
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