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# Linear systems homework?

Here is the information:

Abiman, an exchange student from Sri Lanka, is studying in New Zealand. He phones his friends who are living in Australia, Britain, and China. The rates per minute for these phone calls vary for different countries.

The table below shows the total charges for the months of September, October and November and the length of time of the phone calls in minutes.

(((((((((The table is attached as a photo)))))))

1. Write a system of three linear equations using the information in the table

2. Solve this set of equations to find the calling rates per minute for each of these countries.

Please link this to the context and use geometrical reasoning when needed to help me understand. Thank you :) ### 1 Answer

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Beginning equations:

90a + 120b + 180c = 15 (Eq. 1)

70a + 100b + 120c = 11 (Eq. 2)

50a + 110b + 150c = 12.10 (Eq. 3)

1. Divide all 3 equations by 5.

18a + 24b + 36c = 3 (new eq. 1)

14a + 20b + 24c = 2.2 (new eq. 2 )

10a + 22b + 30c = 2.42 (new eq. 3)

2. Subtract (3/2) times equation 1 from equation 2

eq. 2

14a +20b +24c- (2/3)*(18a+ 24b + 36c) = 2.2 - (2/3)*3

(14 - 12)a + (20-16)b + 0c = 2.2 - 2

2a + 4b + 0c = 0.2 (new eq. 2 )

so now you have

18a + 24b + 36c = 3 (new eq. 1)

2a + 4b = 0.2 (new eq. 2 )

10a + 22b + 30c = 2.42 (new eq. 3)

3. Subtract 5 times equation 2 from equation 3

10a + 20b + 30c - 5*(2a + 4b) = 2.42 - 5*0.2

(10-10)a + (22-20)b + 30c = 1.42

2b + 30c = 1.42 (new eq.3 )

so now you have

18a + 24b + 36c = 3 (new eq. 1)

2a + 4b = 0.2 (new eq. 2 )

2b + 30c = 1.42 (new eq. 3)

4. Subtract 12 times equation 3 from equation 1

18a + 24b + 36c - 12(2b + 30c) = 3- 5*1.42

18a + (24-24)b + (36- 360) = 14.04

18a -324c = 14.04 (new eq. 3 )

so now you have

18a -324c = -14.04 (new eq. 1)

2a + 4b = 0.2 (new eq. 2 )

2b + 30c = 1.42 (new eq. 3)

5. Subtract (1/9) times equation 1 from equation 2

2a +4b - (1/9)(18a -324c) = 0.2 - (1/9) (-14.04)

(2-2)a + 4b + 36c = 1.76 (new eq. 2)

4b + 36c = 1.76

so now you have

18a -324c = -14.04 (new eq. 1)

4b + 36c = 0.2 (new eq. 2 )

2b + 30c = 1.42 (new eq. 3)

6. Subtract (6/5) times equation 3 from equation 2

4b + 36c - (6/5)(2b + 30c) = 0.2 - (5/6)1.42

(4- 12/5) b + (36-36)c = 0.056

1.6b = 0.056 (new eq. 2)

so now you have

18a -324c = -14.04 (new eq. 1)

1.6b = 0.056 (new eq. 2 )

2b + 30c = 1.42 (new eq. 3)

7. Subtract 1.25 times equation 2 from equation 3

(new eq. 3)

2b + 30c - (1.25)(1.6b) = 1.42 - (1.25)(0.056)

30c = 1.35

so now you have

18a -324c = -14.04 (new eq. 1)

1.6b = 0.056 (new eq. 2 )

30c = 1.35 (new eq. 3)

8. Divide equation 3 by 30

30c/30 = 1.35/30

c = 1.35/30 =

c = 0.045 (4.5 cents per minutes)

9. Divide equation 2 by 1.6

1.6b/1.6= 0.056/1.6

b= 0.035 (3.5 cents per minute )

so now you have

18a -324c = 14.04 (new eq.1 )

b = 0.035 (new eq. 2)

c = 0.045 (new eq. 3 )

10. Add 324 times equation 3 to equation 1

new eq.1

18a -324c + 324(c) = -14.04 + 324(0.045)

18a = 0.54 (new eq. 1 )

11. Divide equation 1 by 18.

a = 0.54/18

a = 0.03

so now you have

a = 0.03 (3 cents per minute)

b = 0.035 (3.5 cents per minute)

c = 0.045 (4.5 cents per minute)

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