MATH 1051 Midterms

Question 1

Determine equations for the following straight line:

The line that goes through the point (1, -2) and is parallel to 3x+4y=6

Answer 1

Equation for a straight line: y-b = t (x-a)

Parallel lines: have the same slope

Perpendicular lines: have slopes that are negative reciprocals; -5/6 and 6/5

If there is an equation solve for Y and pull slope from equation

Question 2

Determine a Linear Equation from Real-World Info

Answer 2

we want y= f(x)

with y = first bolded phrase and x = second bolded phrase.

Two (x,y) points: based on world problem

Final equation: y- b = t (x-a)

Question 3

Determine just by looking the number of points on intersection of each of the following systems of equations and briefly describe the geometry of the system: 3-D lines

Use the word planes in your geometric description

Answer 3

If multiple does not work for coefficients: Planes are neither the same nor parallel; Cannot tell just by looking

If multiple works for x,y,z, and constant: Planes _ and _ are the same and the other is not parallel; Infinitely many pts. of int.

If multiple works for all he coefficients but not constant: Planes _ and _ are parallel; Zero pts. of int.

Parallel planes supersedes all

Attack by 1:2, 1:3, 2:3

Question 3

Determine just by looking the number of points of intersection and briefly describe the Geometry of the System: 2 -D lines

Use the word lines in your geometric description

Answer 3

does not work for both coefficients: One point of intersection; the lines have different slopes

works for both coefficients and constant: the lines are the same; infinitely many points of intersection

works for both coefficients but not constant: the lines are parallel; zero points of intersection

ex. 2x-3y=5
-4x+6y=10

?(2)= -4
-2 works for x and y coefficient but not for constant so zero pts of int

Question 4

Gauss-Jordan: If you produce a row with all zeros to the left of the bar and a no-zero number to the right, you must (inconsistent system)

Answer 4

Complete the pivot and stop, and write that the system has zero points of intersection

Question 5

Gauss-Jordan: If you produce a row of all zeros (dependent system)

Answer 5

you must continue pivoting until you cannot pivot anymore and if your system is not inconsistent you must depict a general from for the infinitely many pts. of int.

Determining what are points of intersection:

are points of intersection;
are not.

Question 6

Matrix Multiplication

Answer 6

multiple row by column

Question 7

Inverse matrices

Answer 7

Five steps sometimes only need four

1. Find the matrix that lines up with the given equations and find the constant
2. rewrite equations as matrices form
3. multiple by inverse matrix on both sides
4. solve and find what they multiply to
5. write the point of intersection

Finding # on right of equal sign:
Use constant from the GIVEN pair of lines to find what you multiple to get the given lines you made

Question 8

Determine the 3-D plane whose point of intersection would solve the following:

Answer 8

x: find variable from problem use context of last sentence
y:
z:

Give EQUATIONS based on bullet points

Question 9

LPP problems for inequality, at least indicates

Answer 9

greater than or equal to sign

Question 10

Simplex Algorithm

Answer 10

constraints must be less than or equal to

Question 11

Basic simplex algorithm

Answer 11

selecting pivot entry:
1. select most (-) # in test region; finds pivot column
2. pivot entry. must be above horizontal bar and (+); dismiss (-) or 0
3. if positives use least c/p ratio (lowest fraction)

if two negatives entries; choose leftmost
if no positive entries above bar; LPP has no solution
if two w/ the same least ratio; choose topmost

Question 12

Full simplex algorithm

Answer 12

special pivot entry row: special pivot entry row is the one w/ the most negative entry in the special test region
if two most negative entries, choose topmost

special pivot entry is the most negative entry in that row and to the left of the vertical bar
if two most negative entries choose leftmost

if no negative entries, LPP has no solution

Question 13

Geometric LPP test points

Answer 13

goes through origin (1,0)
doesn’t go through origin (0,0)

if inequality is valid point in towards (0,0)

Question 14

Geometric LPP with a shape

Answer 14

step up constraint to equals, if point satisfies that means it goes through the point

the set up constraint to inequality if it satisfies, it cuts the points that work and the ones that don’t

Question 15

Listing strategies

Answer 15

includes exactly: pick & place & fill

giving exact place: pick & fill

uses word and: pick & pick & fill

uses for or: words that begin with/ even ( exact place + words that end with/even (exact place) - word that were counted twice (pick even for front, then fill)

uses word includes: hidden or/at most; words w/ zero evens + words w/ exactly one even (includes exactly

Question 16

Probabilities

Answer 16

The number of n-letter passwords that can be formed from m letters, allowing repeats is m^n

the number of n-letter passwords that can be formed from m letters, without repeats is Pmn

Question 17

Meshing

Answer 17

The number of n-letter subsets that can be formed from m letters is Cmn

in reference to more than one number and deals with placement/order

ex. exactly three vowels; 3 letter all vowels, 12 letters no vowels, mesh

at most: 15 letters no vowels, exactly one vowel (includes exactly), exactly two vowels (meshing)