Download Problem Set #2

Problem Set #2
Note: Use RREF on your calculator, unless you need to do by hand, as in #1 and #3.
1. Solve the system of linear equations by row reducing the augmented matrix by hand
to convert it to row-reduced echelon form. Show all steps and label each elementary
row operation.
3x  7 y  z  11
2x  y  2z  9
 x  2 y  2 z  4
2. Solve each of the following systems of equations. For each system, give all solutions,
or explain why none exist. If there are infinitely many solutions, give solutions as a
linear combination of column vector(s):
7 x  y  3z  2w  4
(a) 2 x  5 y  z  3w  1
8 x  13 y  9 z  5w  5
2 x  4 y  3z  4w  7
(b)  x  2 y  z  5w  11
4 x  8 y  7 z  2w  5
3. Under what conditions on k does the system
x  3y  7z  5
2x  7 y  6z  3
have
xk yz k
2
(a) a unique solution
(b) infinitely many solutions
(c) no solutions
4. A rotated hyperbola has equation ax 2  bxy  cy 2  d . Find an equation for the
hyperbola if it is known that it has an x intercept at x  2 and the tangent line
equation at x  1 is y 
30
16
x .
7
7
5. Consider the electric circuit shown below. Given that I 1 
15
85
115
, I2 
and I 3 
19
38
38
(amps) determine the resistances R1, R2 and R3 . If there are multiple solutions,
restrict the value of the parameter(s) so that the solution makes physical sense.
R1
B
R3
A
I1
F
I3
10 V
I2
15 V
R2
C
D
E