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Question 13 of 18

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M:N:MC:12

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table attributes columnalign right center left columnspacing 2px end attributes row cell 1 half x minus 1 fourth y end cell equals 5 row cell a x minus 3 y end cell equals 20 end table

In the system of linear equations above, a is a constant. If the system has no solution, what is the value of a ?

Select an Answer

1 half

Correct Answer: 
No

2

Correct Answer: 
No

6

Correct Answer: 
Yes

12

Correct Answer: 
No

View Correct Answer

Choice C is correct. If the system of equations has no solution, the graphs of the equations in the xy-plane are parallel lines. To be parallel, the lines must have the same slope, and this will be true if the expression a x minus 3 y is a multiple of the expression 1 half x minus 1 fourth y. Since minus 3 y equals 12 open parentheses minus 1 fourth y close parentheses comma the expression a x minus 3 y would have to be 12 times the expression 1 half x minus 1 fourth y. This means a x equals 12 open parentheses 1 half x close parentheses comma so a equals 6. The resulting system is 1 half x minus 1 fourth y equals 5 and 6 x minus 3 y equals 20 comma which is equivalent to 6 x minus 3 y equals 60 and 6 x minus 3 y equals 20 comma which has no solution.

Choice A is not the correct answer. This may result from the misconception that if each equation in a system has the same x-coefficient, the system cannot have a solution. But if a equals 1 half comma subtracting the two equations eliminates x and produces a solution to the system.

Choice B is not the correct answer. This may result from trying to make the second equation in the system a multiple of the first by looking at the ratio of the constants on the right sides, 20 over 5 comma and wrongly concluding that the second equation must be 4 times the first, which would give a equals 4 open parentheses 1 half close parentheses comma or a equals 2. But the two equations in a system are multiples only if the system has infinitely many solutions, not if the system has no solution. 

Choice D is not the correct answer. The student may have found the factor, 12, that multiplies the left side of the first equation to yield the left side of the second, but then neglected to find a equals 12 open parentheses 1 half close parentheses comma or a equals 6.

Question Difficulty: 
hard
Objective: 

In addition to solving systems of linear equations that have a solution, students must be familiar with systems that have no solution or an infinite number of solutions.