In the system of linear equations above, is a constant. If the system has no solution, what is the value of
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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 is a multiple of the expression Since the expression would have to be 12 times the expression This means so The resulting system is and which is equivalent to and 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 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, and wrongly concluding that the second equation must be 4 times the first, which would give or 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 or
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.