Consider the following 15 numbers 1, 2, y - 4, 4, 5, x, 6, 7, 8, y, 9, 10, 12, 3x, 20 - The mean o

Consider the following 15 numbers

1, 2, y - 4, 4, 5, x, 6, 7, 8, y, 9, 10, 12, 3x, 20

- The mean of the last 10 numbers is TWICE the mean of the first 10 numbers

- The sum of the last 2 numbers is FIVE times the sum of the first 3 numbers

(i) Calculate the values of x and y

We're given two equations:

1. (x + 6 + 7 + 8 + y + 9 + 10 + 12 + 3x + 20)/10 = 2(1 + 2 + y - 4 + 4 + 5 + x + 6 + 7 + 8 + y)/10
2. 3x - 20 = 5(1 + 2 + y - 4)

Let's evaluate and simplify:

1. (x + 6 + 7 + 8 + y + 9 + 10 + 12 + 3x + 20)/10 = (1 + 2 + y - 4 + 4 + 5 + x + 6 + 7 + 8 + y)/5
   
2. 3x - 20 = 5(y - 1)

Simplify some more:
(x + 6 + 7 + 8 + y + 9 + 10 + 12 + 3x + 20)/10 = (4x + y + 72)/10
(1 + 2 + y - 4 + 4 + 5 + x + 6 + 7 + 8 + y)/5 = (2y + x + 29)/5
5(y - 1) = 5y - 5

So we're left with:

1. (4x + y + 72)/10 = (2y + x + 29)/5
   
2. 3x - 20 = 5y - 5

Cross multiply equations in 1, we have:
5(4x + y + 72) = 10(2y + x + 29)
20x + 5y + 360 = 20y + 10x + 290

We have:

1. 20x + 5y + 360 = 20y + 10x + 290
   
2. 3x - 20 = 5y - 5

Combining like terms:

1. 10x - 15y = -70
2. 3x - 5y = 15

Now we have a system of equations which we can solve any of three ways:

• Substitution Method
• Elimination Method
• Cramer's Rule

No matter which method we choose, we get the same answer:
(x, y) = (-115, -72)