Linear Systems

1. Solve the following systems of equations below.

Answer: (11/2,−1)

Answer: (−71, 17)

Answer: Any pair of the form (5 − 2y, y). These two equations represent
the same line .

2. Give an example of a linear system of equations with no solution in
which at least one of the lines has slope equal to −1/3.

Answer.

3. New vehicles typically depreciate the minute you drive them off the
lot and then continue to depreciate gradually as time goes by. While
this depreciation is typically not quite linear , we will assume that it
it for purposes of this problem. Let n be the number of years passed
since purchase of the vehicle and denote by V_A, V_B, and V_C the values
of vehicles of each of the three models after n years have passed.

(a) Determine three equations relating each of V_A, V_B, and V_C to n.

Answer.

(b) Use your calculator to generate a table that gives the number of
years passed and the values of vehicles of each of the three models
for ten years.
Answer.

(c) Use your calculator to graph the three equations on the same axes
(i.e., graph the system of equations ).

(d) Over what (roughly) time range is a model A vehicle worth more
than either of the other two ? Answer the same questions for model
B and model C vehicles.

Answer A is most valuable after year 11.8095. B is most valuable
between years 6.4 and 11.8095. C is most valuable prior to year
6.4.

(e) Calculate the intersection points of the equations in the system.
Answer. V_A and V_B intersect at the point (11.8095, 11342.9);
V_A and V_C intersect at the point (10.0645, 12651.6); V_B and V_C
intersect at the point (6.4, 18240)

4. Jill leaves Boston at noon, traveling west at a constant speed of 55 miles
per hour; Bob leaves Boston three hours later, traveling at a constant
speed of 60 miles per hour.

(a) If D_J and D_B represent the distances each person has traveled in
t hours, write equations that relate D_J and D_B to t.
Answer. Letting t be the number of hours past 3:00,

(b) If Bob has been on the road for four hours, how long has Jill been
on the road?
Answer. Seven hours

(c) Graph the equations from part (a) on the same set of axes.

(d) Determine where Bob will overtake Jill, that is, determine both
how long it will take and how far West of Boston will they be
when it happens.

Answer. Bob will overtake Jill 33 hours after his departure, when
they are 1980 miles into their trip.

5. Graph (on separate graphs) each of the following inequalities

6. Let

(a) Compute f(−10), f(−3), f(0), f(5), and f(12).
Answer. −37, 4, 4, 1, and −6, respectively.

(b) Graph y = f(x).

7. Let GT_MS be the tax on people in Massachusetts who file under the
category “Married/separate” under the graduated tax plan outlined on
page 179 and x be the taxable income of such people. Let FT_MS be the
tax on such people under a flat tax plan with a flat tax rate of 5.7%.

(a) Use the information in Table 3.5 in the text (bottom of page 167)
to express GT_MS as a function of x.

(b) Make an input-output table for the function GT_MS using incomes
ranging from 0 to $100,000 in $10,000 increments.

(c) Make a graph that shows GT_MS and FT_MS on the same set of
axes, where FT_MS is the flat tax at a 5.7% rate on income.

(d) At what income does one pay the same tax regardless of the tax
plan?
Answer. $ 43,111.80

Extra.

8. Consider the function s(x) defined below

(a) Sketch a graph of this function. This function is sometimes called
a “step” function because of the appearance of its graph.

(b) Modify s(x) to get a function whose graph is a step that is twice
as high as the step in s (x).
Answer. 2s(x) or s(x) + 1

(c) Modify s(x) to get a function whose graph is a step that is two
units further to the right as the step in s(x).
Answer. s(x − 2)

9. If A is a set of numbers, the function

is called the indicator function of A. Explain how your calculator uses
indicator functions in graphing piecemeal functions.

10. Let A and B be sets of functions. Show that

11. Write a formula for the function whose graph is below using indicator
functions.