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Chapter 8: Problem 90

When using the addition or substitution method, how can you tell if a system of linear equations has infinitely many solutions? What is the relationship between the graphs of the two equations?

Short Answer

Expert verified
If the addition or substitution method results in a true statement like 0=0, the system of linear equations has infinitely many solutions. Graphically, the lines representing the equations in the system coincide entirely, meaning they have the same gradient and y-intercept.

Step by step solution

01

Recognizing the System of Equations

When working with a system of equations, solutions can be considered as points where the equations intersect when graphed. So if there are infinitely many solutions, it means that the equations coincide and therefore the graphs overlap completely.
02

Checking Characteristic in Addition or Substitution Method

In both the addition and substitution methods, if it leads to a true statement, such as 0 = 0, it means the original system actually represents the same linear equation and has infinitely many solutions.
03

Considering the graphical representation

In a graphical context, if a system has infinitely many solutions, the lines representing the equations will be the exact same line i.e., they coincide - every point on the line is an intersection. This means that the equations are just re-arrangements or scalings of each other, thus having the same gradient and y-intercept, leading to infinitely many points of intersection.

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Most popular questions from this chapter

A television manufacturer makes rear-projection and plasma televisions. The profit per unit is 125 for the rear-projection televisions and 200 for the plasma televisions. a. Let \(x=\) the number of rear-projection televisions manufactured in a month and let \(y=\) the number of plasma televisions manufactured in a month. Write the objective function that models the total monthly profit. b. The manufacturer is bound by the following constraints: \(\cdot\) Equipment in the factory allows for making at most 450 rear-projection televisions in one month. \(\cdot\) Equipment in the factory allows for making at most 200 plasma televisions in one month. \(\cdot\) The cost to the manufacturer per unit is 600 for the rear-projection televisions and 900 for the plasma televisions. Total monthly costs cannot exceed 360,000. Write a system of three inequalities that models these constraints. c. Graph the system of inequalities in part (b). Use only the first quadrant and its boundary, because \(x\) and \(y\) must both be nonnegative. d. Evaluate the objective function for total monthly profit at each of the five vertices of the graphed region. [The vertices should occur at \((0,0),(0,200),(300,200)\) \((450,100), \text { and }(450,0) .]\) e. Complete the missing portions of this statement: The television manufacturer will make the greatest profit by manufacturing _____ rear- projection televisions each month and _____ \(-\) plasma televisions each month. The maximum monthly profit is $_____.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Even if a linear system has a solution set involving fractions, such as \(\left\\{\left(\frac{8}{11}, \frac{43}{11}\right)\right\\},\) I can use graphs to determine if the solution set is reasonable.

A company that manufactures small canoes has a fixed cost of \(\$ 18,000 .\) It costs \(\$ 20\) to produce each canoe. The selling price is \(\$ 80\) per canoe. (In solving this exercise, let \(x\) represent the number of canoes produced and sold.)

Use the two steps for solving a linear programming problem, given in the box on page \(888,\) to solve the problems. You are about to take a test that contains computation problems worth 6 points each and word problems worth 10 points each. You can do a computation problem in 2 minutes and a word problem in 4 minutes. You have 40 minutes to take the test and may answer no more than 12 problems. Assuming you answer all the problems attempted correctly, how many of each type of problem must you answer to maximize your score? What is the maximum score?

Write a system of equations having {(-2, 7)} as a solution set. (More than one system is possible.)
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