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

Question

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

Accepted Answer

Step 1: Let the three consecutive integers be x = n, y = n+1, z = n+2.

Step 2: Substitute these into the equations:

Equation 1: x - y + z = 2a
=> n - (n+1) + (n+2) = 2a
=> n - n - 1 + n + 2 = 2a
=> (n - n + n) + (-1 + 2) = 2a
=> n + 1 = 2a

Equation 2: x + 4y - 2z = 3(4 - a)
=> n + 4(n+1) - 2(n+2) = 12 - 3a
=> n + 4n + 4 - 2n - 4 = 12 - 3a
=> (n + 4n - 2n) + (4 - 4) = 12 - 3a
=> 3n = 12 - 3a
=> n = 4 - a

Equation 3: 2x - 3y + 4z = 6 - 2a
=> 2n - 3(n+1) + 4(n+2) = 6 - 2a
=> 2n - 3n - 3 + 4n + 8 = 6 - 2a
=> (2n - 3n + 4n) + (-3 + 8) = 6 - 2a
=> 3n + 5 = 6 - 2a

Step 3: From equation 2, n = 4 - a.

Step 4: Substitute n into equation 1:

n + 1 = 2a
=> (4 - a) + 1 = 2a
=> 5 - a = 2a
=> 5 = 3a
=> a = 5/3 (not an integer, so check equation 3)

Step 5: Substitute n into equation 3:

3n + 5 = 6 - 2a
=> 3(4 - a) + 5 = 6 - 2a
=> 12 - 3a + 5 = 6 - 2a
=> 17 - 3a = 6 - 2a
=> 17 - 6 = -2a + 3a
=> 11 = a

Step 6: We have two different values for a (5/3 and 11), which is inconsistent. This means the assumption that x, y, z are consecutive integers in the order n, n+1, n+2 is incorrect.

Step 7: Try the order x = n, y = n-1, z = n-2 (consecutive integers in decreasing order):

Equation 1:

x - y + z = 2a
=> n - (n-1) + (n-2) = 2a
=> n - n + 1 + n - 2 = 2a
=> (n - n + n) + (1 - 2) = 2a
=> n - 1 = 2a

Equation 2:

x + 4y - 2z = 3(4 - a)
=> n + 4(n-1) - 2(n-2) = 12 - 3a
=> n + 4n - 4 - 2n + 4 = 12 - 3a
=> (n + 4n - 2n) + (-4 + 4) = 12 - 3a
=> 3n = 12 - 3a
=> n = 4 - a

Equation 3:

2x - 3y + 4z = 6 - 2a
=> 2n - 3(n-1) + 4(n-2) = 6 - 2a
=> 2n - 3n + 3 + 4n - 8 = 6 - 2a
=> (2n - 3n + 4n) + (3 - 8) = 6 - 2a
=> 3n - 5 = 6 - 2a

Step 8: Substitute n = 4 - a into equation 1:

n - 1 = 2a
=> (4 - a) - 1 = 2a
=> 3 - a = 2a
=> 3 = 3a
=> a = 1

Step 9: Substitute a = 1 into equation 3:

3n - 5 = 6 - 2a
=> 3(4 - 1) - 5 = 6 - 2(1)
=> 3(3) - 5 = 6 - 2
=> 9 - 5 = 4
=> 4 = 4 (True)

Step 10: So, a = 1 satisfies all equations with x, y, z as consecutive integers in decreasing order: x = n = 4 - a = 3, y = n - 1 = 2, z = n - 2 = 1.

Step 11: Check the first equation with these values:

x - y + z = 3 - 2 + 1 = 2
2a = 2 * 1 = 2 (True)

Step 12: Check the second equation:

x + 4y - 2z = 3 + 8 - 2 = 9
3(4 - a) = 3(4 - 1) = 9 (True)

Step 13: Check the third equation:

2x - 3y + 4z = 6 - 6 + 4 = 6 - 2a
=> 6 - 6 + 4 = 6 - 2(1)
=> 4 = 4 (True)

Therefore, the value of a is 1.

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