Quiz 1 solutions - Quiz 1 Solutions 1(a Adding R1 to 3R2 we...

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Quiz 1 Solutions 1 . (a) Adding - R 1 to 3 R 2 , we get the equation ( 1 - 3 a ) y = - 5 . Thus the system of equations fails to have a unique solution if and only if 1 - 3 a = 0 , i.e. if and only if a = 1 3 . (b) If a = 1 3 , then ( 1 - 3 a ) y = - 5 becomes 0 = 5 , so that the system of equations is inconsistent. Therefore no values of x and y satisfy both equations, so the solution set is the empty set. (c) If the right hand value - 2 is replaced by 1 3, the two equations are then scalar multiples of each other, so they represent the same line in the plane, namely y = - 3 x + 1 , the solution set. Note: since x,y Q , we should be careful and say the solution set is the set {( x, - 3 x + 1 ) such that x Q } . (d) The solution set is a line, which has dimension 1

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