1 Math-132 Fall 2017 Final Exam Review Section 1.1 #1) Find the equation of a line with slope 1 2 that passes through the point (3, 1), and write the equation in slope-intercept form. [answer: = − ] #2) Find the equation of the line that connects the points (−6, 2) (−1, −2), and write the equation in slope-intercept form. [answer: =− − ] #3) Find the equation of the line that passes through the points (2, −1) (0, 5), and write the equation in slope-intercept form. [: = − + ] Section 1.2 #4) Determine the solution of each of the following systems and put your solution in point form (, ). a) { 4 − 2 = 16 4 + 2 = 0 [answer= (, −)] b) { 4 + 3 = 2 2 − = −1 [answer= (− , )] #5) A retiree needs $10,000 per year in supplementary income. He has $150,000 to invest and can invest and can invest in AA bonds at 10% annual interest or in Savings and Loans certificates of 5% interest per year. How much money should be invested in each so that he realizes exactly $10,000 in extra income per year? [answer= $, & $, ]
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Math-132
Fall 2017 Final Exam Review
Section 1.1
#1) Find the equation of a line with slope 1
2 that passes through the point (3, 1), and write the equation
in slope-intercept form.
[answer: 𝒚 =𝟏
𝟐𝒙 −
𝟏
𝟐]
#2) Find the equation of the line that connects the points (−6, 2) 𝑎𝑛𝑑 (−1,−2), and write the
equation in slope-intercept form.
[answer: 𝒚 = −𝟒
𝟓𝒙 −
𝟏𝟖
𝟓]
#3) Find the equation of the line that passes through the points (2, −1) 𝑎𝑛𝑑 (0, 5), and write the equation in slope-intercept form. [𝒂𝒏𝒔𝒘𝒆𝒓: 𝒚 = −𝟑𝒙 + 𝟓 ] Section 1.2 #4) Determine the solution of each of the following systems and put your solution in point form (𝑥, 𝑦).
a) {4𝑥 − 2𝑦 = 164𝑥 + 2𝑦 = 0
[answer= (𝟐,−𝟒)]
b) {4𝑥 + 3𝑦 = 22𝑥 − 𝑦 = −1
[answer= (−𝟏
𝟏𝟎,𝟒
𝟓)]
#5) A retiree needs $10,000 per year in supplementary income. He has $150,000 to invest and can invest
and can invest in AA bonds at 10% annual interest or in Savings and Loans certificates of 5% interest per
year. How much money should be invested in each so that he realizes exactly $10,000 in extra income
Section 1.3 #6) A manufacturer produces gameday pennants at a cost of $0.85 per item and sells them for $0.95 per
item. The daily operational overhead is $350.
a) What is the daily cost function?
[𝒂𝒏𝒔𝒘𝒆𝒓: 𝑪(𝒙) = 𝟎. 𝟖𝟓𝒙 + 𝟑𝟓𝟎]
b) What is the daily revenue function?
[𝒂𝒏𝒔𝒘𝒆𝒓: 𝑹(𝒙) = 𝟎. 𝟗𝟓𝒙]
c) How many pennants should the manufacturer sell per day in order to break-even?
[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟑𝟓𝟎𝟎]
#7) The supply and demand equations for sugar have been estimated to be given by the equations:
𝑆 = 0.6𝑝 + 0.5
𝐷 = −0.5𝑝 + 2.7
Where 𝑝 is the price in dollars per pound and 𝑆 and 𝐷 are in millions of pounds.
a) Find the market price.
[𝒂𝒏𝒔𝒘𝒆𝒓: 𝒑 = $𝟐 ]
b) What quantity of supply is required at this market price?
[𝒂𝒏𝒔𝒘𝒆𝒓: 𝑺 ≅ 𝟏. 𝟕 𝒖𝒏𝒊𝒕𝒔]
#8) It costs a cellphone parts manufacturer $12 per item to produce a certain electronic component. There is a fixed cost of $594 per month for the manufacturer to produce the component. The company sells the components for $18 a piece.
a) What is the total cost function per month for producing the electronic component? [𝒂𝒏𝒔𝒘𝒆𝒓: 𝑪(𝒙) = 𝟏𝟐𝒙 + 𝟓𝟗𝟒 ]
b) What is the total revenue function? [𝒂𝒏𝒔𝒘𝒆𝒓: 𝑹(𝒙) = 𝟏𝟖𝒙 ]
c) After selling how many of the electronic components per month will the manufacturer break-even? [𝒂𝒏𝒔𝒘𝒆𝒓: 𝒙 = 𝟗𝟗 ]
Section 2.1 #9) Using the method of elimination, solve each of the following systems and put your solution in the form (𝑥, 𝑦, 𝑧).
Section 2.2 & 2.3 #10) The following augmented matrix is already in Row Echelon Form (REF). Find the solution of the system and give the answer in the form (𝑥1, 𝑥2, 𝑥3).
(1 7 60 1 00 0 1
|7−11)
[answer= (𝟖,−𝟏, 𝟏)]
#11) Given the following augmented matrix, use row operations (i.e., the Gaussian elimination) to put the matrix into Row Echelon Form (REF). Then, determine the solution to the system and put your answer in the form (𝑥1, 𝑥2).
(3 21 − 2
|142)
[answer= (𝟒, 𝟏)]
#12) Given the following augmented matrix, use row operations (i.e., the Gaussian elimination) to put the matrix into Row Echelon Form (REF). Then, determine the solution to the system and put your answer in the form (𝑥1, 𝑥2, 𝑥3).
(1 1 13 2 − 13 1 2
|6411)
[answer= (𝟏, 𝟐, 𝟑)]
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Section 3.2
#13) Find the product of the following matrices:
a) [4 − 2 3] [033]
[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟑 ]
b) [1 9] [5 09 − 5
]
[𝒂𝒏𝒔𝒘𝒆𝒓: [𝟖𝟔 − 𝟒𝟓] ]
c) [1 − 4 52 0 10
] [0 − 51 04 − 10
]
[𝒂𝒏𝒔𝒘𝒆𝒓: [𝟏𝟔 − 𝟓𝟓𝟒𝟎 − 𝟏𝟏𝟎
]]
d) [8 010 − 811 − 1
] [8 19 − 8
]
[𝒂𝒏𝒔𝒘𝒆𝒓: [𝟔𝟒 𝟖𝟖 𝟕𝟒𝟕𝟗 𝟏𝟗
]]
Section 3.3
#14) Using elementary row operations (not the determinant method) to find the inverse of the
following matrices:
a) 𝐴 = [8 164 9
]
[𝒂𝒏𝒔𝒘𝒆𝒓: 𝑨−𝟏 = [𝟗
𝟖 −
𝟏
𝟖
−𝟖 𝟏] ]
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b) 𝐴 = [1 − 1 0−3 4 02 0 1
]
[𝒂𝒏𝒔𝒘𝒆𝒓: 𝑨−𝟏 = [4 1 03 1 0−8 − 2 1
]]
c) #8) 𝐴 = [1 − 1 0−6 7 08 0 1
]
[𝒂𝒏𝒔𝒘𝒆𝒓: 𝑨−𝟏 = [7 1 06 1 0−56 − 8 1
]]
Sections 4.1 & 4.2
#15) The given figure illustrates the graph of the set of feasible points of a linear system of inequalities.
Find the minimum and maximum values of the objective function 𝑧 = 3𝑥 + 11𝑦.
[𝒂𝒏𝒔𝒘𝒆𝒓: 𝒎𝒊𝒏𝒊𝒎𝒖𝒎 = 𝟏. 𝟐 ]
[𝒂𝒏𝒔𝒘𝒆𝒓: 𝒎𝒂𝒙𝒊𝒎𝒖𝒎 = 𝟐𝟏 ]
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#16) Using the geometrical method, find a graphical solution to each of the following linear systems of
inequalities. Then, find the coordinates of the corner points for each solution region.
a)
{
𝑥 ≤ 10𝑦 ≤ 8
4𝑥 + 3𝑦 ≥ 12𝑥 ≥ 0𝑦 ≥ 0
[𝒂𝒏𝒔𝒘𝒆𝒓: (𝟑, 𝟎) (𝟏𝟎, 𝟎) (𝟏𝟎, 𝟖) (𝟎, 𝟖) (𝟎, 𝟒) ]
#17) Use the Geometric method to maximize the objective function 𝑧 = 7𝑥 + 5𝑦 subject to the