Math 141 WIR2_Dr. Rosanna Pearlstein
1.
1.Q, 2.1, 2.2
For the following quadratic equations, do the following:
a) Determine whether the graph opens up or down.
b) Find the vertex of the parabola.
c) Find the maximum or minimum values.
d) Find the x-intercepts, if any.
I. 𝒚 = 𝟐𝟓𝒙𝟐 − 𝟏𝟎𝒙 + 𝟏
II. 𝒚 = 𝟕(𝒙 + 𝟒)𝟐 − 𝟏𝟎
2.
If the demand price of an item is given by 𝑝(𝑥) = 180 − 3𝑥 (where x is the number of items) find the revenue function.
a) What number of items produced yields maximum revenue?
b) If the cost is 𝑐(𝑥) = 171𝑥 − 12, find the equilibrium point.
3.
For each linear system below, use the Gauss-Jordan elimination method to determine the solution(s) or lack thereof.
A.
B.
C.
3
𝑥 − 4𝑦 = −6
4
4𝑥 − 2𝑦 = −1
2
3
𝑥+
𝑦=2
5
10
{
4
20
{ 3𝑥 + 𝑦 = 3
5
2
{2 𝑥 + 3 𝑦 = 22
−2𝑥 + 𝑦 = 1
4.
(From our textbook, Finite Mathematics by Tan, 2.1 #28). Box Office Receipts. A theater has a seating capacity of 900 and
charges $4 for children, $6 for students, and $8 for adults. At a certain screening with full attendance, there were half as
many adults as children and students combined. The receipts totaled $5600. How many children attended the show?
5.
Which of the matrices below is in row-reduced form?
A.
B.
1 2
[0 0
0 0
6.
0 6
1 |5]
0 1
C.
1
[0
0
0
1
1
0 −3
0 |4]
−1 1
1
[0
0
2
0
0
3 −7
1| 0 ]
0 1
Pivot each of the two matrices below about the boxed element.
A.
B.
1
[0
0
2
−6
5
8 6
9 |5]
−1 3
3
[3
2
6
1
4
9 −3
0 |4]
−1 1
7.
Solve the two linear systems below, using the Gauss-Jordan elimination method as performed by “rref” on your calculator.
𝟐𝒙 = 𝒚
𝒚 − 𝟑𝒙 + 𝟐𝒛 = 𝟖
{𝟖𝒙 + 𝟑𝒚 = 𝟗 + 𝟐𝒛
{ 𝟑𝒚 = 𝟏 + 𝟐𝒙
𝟐𝒛 = 𝟑 + 𝟐𝒚 − 𝒙
𝟒𝒙 = 𝟗 − 𝟓𝒚
8.
Solve the linear systems below. Express the solutions using parameters.
I.
II.
𝟖𝒙 − 𝟒𝒚 − 𝟐𝒛 = −𝟒
{ 𝒙+𝒚+𝒛= 𝟏
𝟏𝟎𝒙 − 𝟐𝒚 = −𝟐
{
−𝟑𝒙 + 𝟐𝒚 − 𝒛 + 𝟒𝒘 = 𝟏
𝒙 + 𝒚 − 𝟓𝒛 − 𝒘 = 𝟐
9.
Determine the value of k for which systems A and B below have infinitely many solutions and no solution respectively.
A.
B.
4𝑥 − 2𝑦 = −1
{
−5𝑥 + 𝑘𝑦 =
5
4
4𝑥 − 𝑘𝑦 = −1
{
−2𝑥 + 3𝑦 = 1
10. Comment on the following statements: are they true or false? Why?
(a) A linear system with the same number of equations as unknowns always has a solution.
(b)
A linear system with more equations than the number of unknowns never has a solution.
(c)
A (2×2) linear system of two lines in the plane will have (at least) one solution if the two lines are not parallel.
(d)
A (3×3) linear system of three planes in space will have a solution if the three planes are not parallel.
(e)
A linear system with more variables than equations always has a solution.
(f)
A linear system with more variables than equations either has no solution or it has infinitely many solutions.