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Chapter 2: Problem 30

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If at least two of the three lines represented by a system composed of three linear equations in two variables are parallel, then the system has no solution.

Short Answer

Expert verified
The statement is true. If at least two of the three lines represented by a system composed of three linear equations in two variables are parallel, then the system has no solution. This is because parallel lines with different y-intercepts never intersect, and hence there is no common point of intersection for all the lines in the system.

Step by step solution

01

Recall properties of parallel lines

Parallel lines have the same slope but different y-intercepts. They never intersect each other. This is important because, in a system of equations, the solution is found at the point of intersection of the lines.
02

Consider a system of three linear equations

Let's consider a system of three linear equations in two variables, \(x\) and \(y\), represented by the equations \[a_i x + b_i y = c_i ,\, i = 1, 2, 3,\]
03

Analyze the condition given

According to the given condition, at least two of these three lines must be parallel. Without loss of generality, let the first two lines be parallel. This means their slopes are equal, that is, \(\frac{-a_1}{b_1} = \frac{-a_2}{b_2}\). But they have different y-intercepts, i.e., \(\frac{c_1}{b_1} \neq \frac{c_2}{b_2}\).
04

Evaluate the possibility of finding a solution

Since the first two lines are parallel and have different y-intercepts, they will never intersect each other. Therefore, no common solution exists for the first two equations. However, the third line can intersect both of these parallel lines at distinct points or can be parallel to them but have a different y-intercept. In either case, the system will not have a unique solution. Thus, the given statement is true. If at least two of the three lines represented by a system composed of three linear equations in two variables are parallel, then the system has no solution, because the parallel lines cannot intersect, and hence there is no common point of intersection for all of them.

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Most popular questions from this chapter

The problems in exercise correspond to those in exercises 15-27, Section 2.1. Use the results of your previous work to help you solve these problems. Lawnco produces three grades of commercial fertilizers. A 100 -lb bag of grade-A fertilizer contains \(18 \mathrm{lb}\) of nitrogen, \(4 \mathrm{lb}\) of phosphate, and \(5 \mathrm{lb}\) of potassium. A \(100-\mathrm{lb}\) bag of grade-B fertilizer contains \(20 \mathrm{lb}\) of nitrogen and \(4 \mathrm{lb}\) each of phosphate and potassium. A 100-lb bag of grade-C fertilizer contains \(24 \mathrm{lb}\) of nitrogen, \(3 \mathrm{lb}\) of phosphate, and \(6 \mathrm{lb}\) of potassium. How many 100 -lb bags of each of the three grades of fertilizers should Lawnco produce if \(26,400 \mathrm{lb}\) of nitrogen, \(4900 \mathrm{lb}\) of phosphate, and \(6200 \mathrm{lb}\) of potassium are available and all the nutrients are used?

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