2x + y = C [1]
3y + z = C [2]
x - 4z = C [3]
PART 1:
Rearrange [3] like so, to get an expression for x:
x = 4z + C
Sub' expression equivalent to x into [1] and rearrange to get an expression for C:
2(4z + C) + y = C
8z + 2C + y = C
8z + C + y = 0
C = -8z - y [4]
This equation will be useful later on so I will let it be [4]
PART 2:
We will also need this rearrangement (can be found by rearranging [4]):
8z + C = -y
We need to take out x from the left side like so:
x = 4z + C
So:
8z + C = x + 4z
And so:
-y = 4z + x
y = -4z - x [5]
This one is also essential so I will let it be [5]
PART 3:
Sub' expression for C from [4] into [2] and rearrange to get an expression for z like so:
-8z - y = 3y + z
9z = -4y
z = -4y/9 [6]
PART 4:
Sub' expression for y from [5] into [2] and simplify like so:
3(-4z - x) + z = C
-12z - 3x + z = C
-11z - 3x = C
PART 5:
Now, sub' in expression for C from [3] into the equation just above and rearrange to express in terms of z:
-11z - 3x = x - 4z
7z = -4x
z = -4/7x [7]
PART 6:
Sub' in expression for z from [6] into [7] and simplify:
-4/7x = -4/9y
-36x = -28y
-9x = -7y
9x = 7y
From the equation just above; we know 9 and 7 must both multiply by two integers to give the same number;
In order to find the two integers (x and y), we first need to find the LCM (lowest common multiple) of the two so:
Multiples of 9: 9, 18, 27, 36, 45, 54, 63, ...
63 is a the LCM, none of the multiples of 9 are also multiples of 7;
All we have to do know is set 9x and 7y equal to 63 and solve for x and y:
9x = 63
x = 7
7y = 63
y = 9
PART 7:
Now, to get z, we just sub' y = 9 or x = 7 into equations [6] and [7] respectively;
It is good practice to double check answers when you can so I would sub' both y = 9 and x = 7 into their respective equations to see if I get the same value (you should get the same value as equations [6] and [7] are both equivalent to z) and hence confirm whether my working is correct;
So:
z = -4/9y
z = -4/9(9)
z = -4
z =-4/7x
z = -4/7(7)
z = -4
As you can, see z = -4 regardless of whether I used the x in the x equation or the y in the y equation.
So, the values of x, y and z are:
x = 7
y = 9
z = -4
And the value of C, therefore, is 23
PART 8:
You should always check your values as I mentioned before so lets do that:
For [1]:
2(7) + 9 = 23
For [2]:
3(9) - 4 = 23
For [3]:
7 - 4(-4) = 23
As you can see, the integer values of x, y and z that I have found can all be subbed in to any of the equations in their respective variables and you will get the same number, 23.
I am quite sure this is the smallest value of C you would find as I did use the lowest common multiple and so I would expect I did get the smallest possible C value.
