Final answer:
The three even numbers that sum up to 162 are 36, 54, and 72. The smallest number is half of the largest, and the middle number is three-fourths of the largest.
Step-by-step explanation:
The question involves solving a system of linear equations to find three even numbers that add up to 162. Let's denote the largest number as L, the smallest as S, and the middle number as M. According to the problem, S is half of L (S = L/2) and M is three-fourths of L (M = 3L/4). The sum of the three is given as S + M + L = 162.
Substituting the values of 'x' and 'y' in the first equation and simplifying, we get (1/2)z + (3/4)z + z = 162. Combining like terms, we have (13/4)z = 162. To solve for 'z,' we can multiply both sides of the equation by 4/13, giving us z = (4/13) * 162 = 48.
Now that we have the value of 'z,' we can substitute it back into the second and third equations to find the values of 'x' and 'y.' Substituting z = 48 into x = (1/2)z, we get x = (1/2)*48 = 24. Similarly, substituting z = 48 into y = (3/4)z, we get y = (3/4)*48 = 36.
Now, substituting the expressions for S and M in terms of L into the sum equation, we get:
L/2 + 3L/4 + L = 162
Simplifying:
(1/2 + 3/4 + 1)L = 162
(9/4)L = 162
L = 162 * (4/9)
L = 72
So, the largest number, L, is 72. Now we find S and M:
S = L/2 = 72/2 = 36
M = 3L/4 = 3*72/4 = 54
Thus, the three even numbers are 36, 54, and 72.