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# Functions and modules

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Functions and Modules Think of a mathematical function that represents something from your own life. For example, suppose the number of beers you drink depends on the number of football games you watch. If you drink five beers during every football game, the function would be:
Number of Beers (B) = 5 times Number of Football Games (F), or B = 5F.
But suppose you drink 2 beers each day irrespective of football, but in addition you drink four beers during every football game. In this case:
Number of Beers (B) is 2 + 4 x Number of Football Games, or B = 2 + 4F.
Create a linear equation from your own life and then write a brief paper describing this linear function. It should relate a particular thing to another thing. It could be something about the amount of money you spend, the number of minutes you spend on the phone, number of miles you run each day, number of push ups, etc…
I normally spend 30 minutes of outgoing calls (average) every day. Also, I spend 2 minutes (average) on every incoming call. Suppose the number of incoming calls that I receive is X and the number of minutes that I spend on the phone is N,
Number of Minutes (N)= 30 + 2 times the incoming calls
N= 30 + 2X
1. Solve for X and Y in the following problems using either substitution or elimination methods. Make sure you show all your work so you can get partial credit even if your final answer is wrong.
a. X + Y= 6  Eqn. 1
2X + Y = 8  Eqn. 2
Subs. Eqn. 1 in Eqn. 2,
X + X + Y= 8
X + 6= 8
X= 2
Subs. X = 2 in Eqn. 1,
2 + Y= 6
Y= 4
b. 7X + 3Y = 14  Eqn. 1
5X + 9Y = 10  Eqn. 2
Eqn. 1 * 3  21X + 9Y= 42
Eqn. 25X + 9Y= 10
(-)16X= 32
X= 2
Subs. X = 2 in Eqn. 1,
14 + 3Y= 14
Y= 0
c. 4X + Y = 16 Eqn. 1
2X + 3Y = 24  Eqn. 2
Eqn. 1 4X + Y = 16
Eqn. 2 * 2 4X + 6Y= 48
(-)-5Y= -32
Y= 32 / 5
Subs. Y = 32 / 5 in Eqn. 1,
4X + (32/5)= 16
4X= 48 / 5
X= 12 / 5
d. 12X + Y = 25 Eqn. 1
8X – 2Y = 14  Eqn. 2
Eqn. 1 * 2 24X + 2Y= 50
Eqn. 2 8X – 2Y= 14
(+)32X= 64
X= 2
Subs. X = 2 in Eqn. 1,
24 + Y= 25
Y= 1
2. Suppose Bob owns 2, 000 shares of Company X and 10, 000 shares of Company Y. The total value of Bobs holdings of these two companies is \$340, 000.
Suppose Frank owns 8, 000 shares of Company X and 6, 000 shares of Company Y. The total value of Franks holdings of these two companies is \$340, 000.
a. Write equations for Bob and Franks holdings. Use the variables X and Y to represent the values of shares of Company X and Company Y.
2, 000X + 10, 000Y= 340, 000 Eqn. 1 (Bob’s Holdings)
8, 000X + 6, 000Y= 340, 000 Eqn. 2 (Frank’s Holdings)
b. Solve for the value of a share of Company X and Company Y. Show your work so you can get partial credit even if your final answer is wrong.
2, 000X + 10, 000Y= 340, 000 Eqn. 1 (Bob’s Holdings)
8, 000X + 6, 000Y= 340, 000 Eqn. 2 (Frank’s Holdings)
Eqn. 1 * 4 8, 000X + 40, 000Y= 1, 360, 000
Eqn. 2 8, 000X + 6, 000Y= 340, 000
(-)34, 000Y= 1, 020, 000
Y= \$30
Subs. Y = 30 in Eqn. 1,
2, 000X + 300, 000= 340, 000
2, 000X= 40, 000
X= \$20
3. Solve for X, Y, and Z in the following systems of three equations using either substition or elimination methods:
a. X + 2Y + Z = 6  Eqn. 1
X + Y = 4  Eqn. 2
3X + Y + Z = 8  Eqn. 3
Subs. Eqn. 2 in Eqn. 1,
X + Y + Y + Z= 6
4 + Y + Z= 6
Y + Z= 2 Eqn. 4
Subs. Eqn. 4 in Eqn. 3,
3X + 2= 8
X= 2
Subs. X = 2 in Eqn. 2,
2 + Y= 4
Y= 2
Subs. X = 2 and Y = 2 in Eqn. 1,
2 + 4 + Z= 6
Z= 0
b. 10X + Y + Z = 12  Eqn. 1
8X + 2Y +Z = 11  Eqn. 2
20X – 10Y – 2Z = 8  Eqn. 3
Eqn. 1 10X + Y + Z = 12
Eqn. 2  8X + 2Y +Z = 11
(-)2X – Y= 1 Eqn. 4
Eqn. 1 * 2 20X + 2Y + 2Z = 24
Eqn. 3  20X – 10Y – 2Z = 8
(+)40X – 8Y= 32 Eqn. 5
Eqn. 4 * 8  16X – 8Y= 8
Eqn. 5 40X – 8Y= 32
(-)-24X= -24
X= 1
Subs. X = 1 in Eqn. 4,
2 – Y= 1
Y= 1
Subs. X = Y = 1 in Eqn. 2,
8 + 2 + Z= 11
Z= 1
c. 22X + 5Y + 7Z = 12  Eqn. 1
10X + 3Y + 2Z = 5  Eqn. 2
9X + 2Y + 12Z = 14 Eqn. 3
Eqn. 1 * 2 44X + 10Y + 14Z= 24
Eqn. 2 * 7 70X + 21Y + 14Z= 35
(-)26X + 11Y= 11 Eqn. 4
Eqn. 2 * 6 60X + 18Y + 12Z= 30
Eqn. 3 9X + 2Y + 12Z= 14
(-)51X + 16Y= 16 Eqn. 5
Eqn. 4 * 16  416X + 176Y= 176
Eqn. 5 * 11 561X + 176Y= 176
(-)X= 0
Subs. X = 0 in Eqn. 4,
0 + 11Y= 11
Y= 1
Subs. X = 0 and Y = 1 in Eqn. 1,
0 + 5 + 7Z= 12
Z= 1

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