If the system of the following equations has the value of the variables as three consecutive integers, then the value of a is ____.
x - y + z = 2a
x + 4y - 2z = 3(4 - a)
2x - 3y + 4z = 6 - 2a
Show answer & explanation
Correct answer: C
Explanation
The correct answer is Option 3 i.e. 1.
Given:
⇒ x - y + z = 2a....(1)
⇒ x + 4y - 2z = 3(4 - a)....(2)
⇒ 2x - 3y + 4z = 6 - 2a....(3)
x, y and z are three consecutive numbers
⇒ x = n, y = n + 1, z = n + 2
Putting the values in eq 1
⇒ x - y + z = 2a
⇒ n - (n + 1) + (n + 2) = 2a
⇒ n - n - 1 + n + 2 = 2a
⇒ n + 1 = 2a
⇒ a = (n + 1)/2....(4)
Now, Putting the values in eq 2
⇒ n + 4(n + 1) - 2(n + 2) = 3(4 - a)
⇒ n + 4n + 4 - 2n - 4 = 12 - 3a [a = (n + 1)/2]
⇒ 3n = 12 - 3(n + 1)/2
⇒ 6n = 12 - 3n - 3
⇒ 9n = 9
⇒ n = 1
On putting the value of n in eq 4
⇒ a = (n + 1)/2
⇒ a = (1 + 1)/2
⇒ a = 2/2
⇒ a = 1