Math 101 - CourseworkUnit 1 - Linear Functions, Systems, & Matrices

You may print out the coursework assignments to complete or do all problems on a notebook paper.Most of the Coursework will be completed in class. A selection of the problems will be submittedfor scoring and feedback.

CW - Lesson 1.0CW - Lesson 1.1CW - Lesson 1.2CW - Lesson 1.3CW - Lesson 1.4CW - Lesson 1.5

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CW - Lesson 1.0 Name:FoundationsRectangular Coordinates:

1. The point that is 3 units to the right of the y-axis and 5 units below the x-axis has thecoordinates ( , ).

2. If x is negative and y is positive, the point (x, y) is in Quadrant .

3. Suppose that A = (2, 5) are the coordinates of a point in the xy-plane. Find the coordinatesof the point if A is shifted 3 units to the left and 8 units up.

4. Plot the points (2, 0), (2,−3), (2, 4), (2, 1), and (2,−1). Describe the set of all points of theform (2, y), where y is a real number.

The Distance and Midpoint FormulasLet (x1, y1) and (x2, y2) be two distinct points, then:

5. Find the distance d between the points (−3, 5) and (3, 2). Express the answer as a simplifiedradical and an approximation rounded to two decimal places.

6. A major leagues baseball ’diamond’ is actually a square, 90 feet on a side. What is the distancedirectly from home plate to second base?

7. Wal-Mart Stores, Inc. reported net sales of $204 billion in 2002 and $375 billion in 2008.Use the midpoint formula to estimate the net sales in 2005. How does this compare to thereported figure of $282 billion?

8. Plot the points A = (−1, 8) andM = (2, 3) in the xy-plane. If M is the midpoint of the linesegment AB, what are the coordinates of the point B?

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CW - Lesson 1.1 Name:

1. Let F (C) be the Fahrenheit temperature corresponding to the Celsius temperature C. Averbal description of the function is given by ”To convert from Celsius to Fahrenheit, multiplythe Celsius temperature by 9

5 , then add 32.” Find ways to represent this function

(a) Algebraically (using a formula)

(b) Numerically (let C = -10, 0, 10, 20, 30, 40)

(c) Visually (graph)

2. Which of the following functions have 5 in their domain? Find the value of the function at 5where possible.

(a) f(x) = x2 − 3x (b) g(x) =x− 5

5(c) h(x) =

√x− 10

3. g(x) =1− x

1 + x; find g(−2), g(a− 1), g(−1)

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CW - Lesson 1.2 Name:

1. Plot the points: A (6, -7), B (11, -3), and C (2, -2).

(a) Show that the ∆ABC is a right triangle.

(b) Find the perimeter of ∆ABC.

(c) Find the area of ∆ABC.

(d) Find the equation of the line that passes through point C and is parallel to the linesegment AB.

2. Solve: .1(x− 2) + .03(x− 4) = .02(10)

3. Find the zero of the function: f(x) =1

3(1− x)− x + 1

2+ 2

4. Solve: 5− 4x = 5x− (9 + 9x)

5.2x

x2 − 1=

2

x + 1− 1

x− 1

6. Solve. Write the answer in interval notation.3

2(6x− 3)− 7x ≥ 3− (7− x)

7. Solve for the indicated variable:

(a) Q = C(2 + EF ) for F

(b) W = qt + qkv for q

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CW - Lesson 1.3 Name:Complete these problems on a separate sheet of paper.

1. The cost of renting office space in a downtown office building varies directly with the size ofthe office. A 600-square-foot office rents for $2550 per month. Use a variation model to findthe rent on a 960 square-foot office.

2. A small business buys a computer for $4000. After 4 years the value of the computer isexpected to be $200. For accounting purposes the business uses linear depreciation to assessthe value of the computer at a given time. This means that if V is the value of the computerat time t, then a linear equation is used to relate V and t.

(a) Find a linear equation that relates V and t.

(b) Sketch a graph of this linear equation.

(c) What do the slope and V -intercept of the graph represent?

(d) Find the depreciated value of the computer 3 years from the date of purchase.

3. The manager of a furniture factory finds that the cost of manufacturing chairs depends linearlyon the number of chairs produced. It costs $2200 to make 100 chairs and $4800 to make 300chairs. Find a linear equation in the form y = mx+b that models the cost of making x chairs,then find how much it costs to make 425 chairs.

4. The population of Kansas City, Missouri was 435,000 in 1990 and 482,300 in 2005. If thepopulation is changing linearly, write an equation that gives the population P as a functionof time, t. Using the model determine the expected population in 2020.

5. A baker makes cakes and sells them at county fairs. His initial cost for the LeavenworthCounty fair was $45 to reserve a booth and $25 traveling expenses. He figures that it costs$1.75 to make each cake and he charges $6.50 per cake. Let x represent the number of cakessold.

(a) Express the revenue R as a function x.

(b) Express the cost C as a function of x.

(c) How many cakes will he need to sell before he will make a profit?

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6. A 50 ft long bungee will stretch a certain amount, depending on how much the person doingthe jump weighs. The following table tells how much a bungee cord will stretch for certainweights. The stretch is in addition to the original 50 ft length of the bungee cord.

Weight (lbs) Stretch (ft)

100 80.9

110 86.7

120 92.4

130 98.1

140 103.7

(a) Use the linear regression process to write an equation that will predict the stretch distancefor a given weight:

stretch = × weight +

(b) Use your equation to find the stretch distance for your weight.

(c) If the concrete is 200 ft below the point where the bungee cord is attached, what is theheaviest ”safe” weight for a 6 ft tall jumper? Assume the bungee cord is attached to thejumper’s ankles.

7. Biologists have observed that the chirping rate of crickets of a certain species appear o berelated to temperature. The following table shows the chirping rate for various temperatures.

(a) Make a scatter plot of the data.

(b) Find and graph the regression line that models the data.

(c) Use the linear model in part (b) to estimate the chirping rate at 100◦

Temperature Chirping rate◦F (chirps/min)

50 2055 4660 7965 9170 11375 14080 17385 19890 211

8. A weight-loss clinic keeps records of its clients‘ weights when they first joined and their weight12 months later. The data for ten clients are shown in the table.

(a) Find the regression line for the weight after 12 months in terms of the beginning weight.

(b) Make a scatter plot of the weights, together with the regression line you found in part(a). Does the line seem to fit the data well?

(c) What does the regression line predict that the weight of a 200-lb person will be 12 monthsafter entering the program?

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x = Beginning weight y = Weight after 12 months

212 188161 151165 166142 131170 165181 156165 156172 152312 276302 289

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CW - Lesson 1.4 Name:

1. Solve the the following systems by graphing, substitution or elimination and write the solu-tions as order pairs. If the system is dependent, write the solution in terms of the secondvariable.

(a)

{25x− 75y = 100−10x + 30y = −40

(b)

{a− 30b = −5

−3a + 80b = 5

(c)

2

x+

1

y= 11

3

x− 5

y= 10

(d)

{0.24x + .32y = .952.4x− 1.2y = 5.9

2. Create a system of linear equations in two variables with the solution (−5, 2).

3. Solve the following 3X3 linear systems, or show that that there is no solution, by using theoperations that lead to an equivalent system in triangular form. If the system has infinitelymany solutions, express them in ordered triple form in terms of the third variable.

(a)

3x + y + 5z = 8

x− y + 2z = 22x− y − 2z = −7

(b)

x− 2y + z = 3

2x− 5y + 6z = 72x− 3y − 2z = 5

(c)

x + 2y − z = 1

2x + 3y − 4z = −33x + 6y − 3z = 4

4. A researcher performs an experiment to test a hypothesis that involves the nutrients niacinand retinol. She feeds one group of laboratory rats a daily diet of precisely 32 units of niacinand 22,000 units of retinol. She uses two types of commercial pellet foods. Food A contains0.12 unit of niacin and 1000 units of retinol per gram. Food B contains 0.20 unit of niacinand 50 units of retinol per gram. How many grams of each food does she feed this group ofrats each day?

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CW - Lesson 1.5 Name:

1. Find three ordered triples that are solutions to the linear equation 2x + 4y − 6z = 12.

2. The following matrices result from applying row operations to the matrix: 1 −1 3 41 2 −2 103 −1 5 14

Next to each matrix, write the operation(s) that was(were) used to transform from the pre-ceding matrix. Finally, write the solution to the system as an ordered-triple: ( , , )

Matrix Row operation(s) used. 1 −1 3 40 3 −5 60 2 −4 2

1 −1 3 4

0 2 −4 20 3 −5 6

1 −1 3 4

0 1 −2 10 3 −5 6

1 −1 3 4

0 1 −2 10 0 1 3

3. Solve the following linear systems by using matrix row operations. Show your work on a

separate sheet of paper. (Use the calculator operation RREF only as a check.)

(a)

2x− y + 3z = 3x + 2y − z = 4

−4x + 5y + z = 10(b)

x− 4y + z = 3

y − 3z = 103y − 8z = 24

(c)

x + 3y − 6z = 72x− y + 2z = 0x + y + 2z = −1

4. Use matrix row operations to solve the system. Write the solution as an ordered-triple.x + y − z = 0

x + 2y − 3z = −32x + 3y − 4z = −3

5. David received an inheritance of $50,000. His financial advisor suggests that he invest thisin three mutual funds: a money-market fund, a blue-chip stock fund, and a high-tech stockfund. The advisor estimates that the money-market fund will return 5% over the next year,the blue-chip fund 9%, and the high-tech fund 16% (all simple interest). David wants a totalfirst-year return of $4000 in interest income. To avoid excessive risk, he decides to invest threetimes as much in the money-market fund as in the high-tech stock fund. How much shouldbe invested in each fund?

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6. A hospital dietician must plan a lunch menu that provides 485 calories, 41.5 g of carbohy-drates, and 35 mg of calcium. A 3-oz serving of broiled ground beef contains 245 Cal, 0 gof carbohydrates and 9 mg of calcium. One baked potato contains 145 cal, 34 g of carbo-hydrates, and 8 mg of calcium. A one-cup serving of strawberries contains 45 cal, 10 g ofcarbohydrates, and 21 mg of calcium. How many servings of each are required to provide thedesired nutritional values?

7. At a college production of A Streetcar Named Desire, 400 tickets were sold. The ticket priceswere $8, $10, and $12, and the total income from the ticket sales was $3700. How manytickets of each type were sold if the combined number of $8 and $10 tickets sold was 7 timesthe number of $12 tickets sold.

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