# 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.

Model |
Price |
Initial Depreciation |
Depreciation per year |

A | 25000 | 4800 | 750 |

B | 32500 | 6100 | 1275 |

C | 35000 | 7000 | 1525 |

(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.

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