# C in average number of vehicles per 15 minutes

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c (in average number of vehicles per 15 minutes) through four inter- sections is shown in D C B A x 3 x 2 x 1 x 4 15 30 15 5 20 35 25 50 Now as an augmented matrix rref 1 0 0 1 35 1 1 0 0 10 0 1 1 0 30 0 0 1 1 15 = 1 0 0 1 35 0 1 0 1 45 0 0 1 1 15 0 0 0 0 0 . Consequently, x 4 is a free variable, say x 4 = t , and x 2 = 45 + t . 2. x 2 = 50 + t 3. x 2 = 40 + t 4. x 2 = 45 + t correct 5. x 2 = 35 + t Explanation: At each intersection flow in equals flow out. Thus x 1 + 50 = x 4 + 15 , x 1 + 25 = x 2 + 35 , x 2 + 20 + 5 + 30 = x 3 + 25 , x 3 + 50 = x 4 + 15 + 20 , which after simplification becomes x 1 x 4 = 35 , x 1 x 2 = 10 , x 2 x 3 = 30 , x 3 x 4 = 15 .
Find all possible values for x 2 in terms of a free variable t . 1. x 2 = 55 + t 009 10.0 points If the equation A x = 0 has the trivial solu- tion, the columns of the matrix A are linearly independent. True or False?
Version 015 – EXAM01 – gilbert – (56780) 5 1. FALSE correct 2. TRUE Explanation: The trivial solution x = 0 is always a so- lution of the homogeneous equation A x = 0 . The columns of a matrix A are linearly inde- pendent if the equation A x = 0 has only the trivial solution. Consequently, the statement is FALSE . 010 10.0 points A basic variable in a linear system corre- sponds to a pivot column in the coe cient matrix for the linear system. True or False? 1 a 5 b + c = 26 . 011 10.0 points Through any three non-collinear points A, B, C in the plane there passes a unique circle. Use the fact that the general equation of a circle is 1. center = (2 , 4) correct 2. center = ( 4 , 8) 3. center = (4 , 8) 4. center = ( 2 , 4) 5. center = ( 4 , 2) 6. center = (8 , 4) Explanation: The circle x 2 + y 2 + ax + by + c = 0 passes through A = ( 1 , 5) when a, b, c satisfy the equation ( 1) 2 + ( 5) 2 a 5 b + c = 0 , i.e. , when
x 2 + y 2 + ax + by + c = 0 to determine the center of this circle when A = ( 1 , 5) , B = ( 1 , 3) , C = (3 , 7) . ( x 2) 2 + ( y + 4) 2 = 10 . Consequently, the circle has center = (2 , 4) .
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