Find the values of x, y and z, so as to satisfy the equations: x + y + z = 12; x + y – z = 6; x – y + z = 4.

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Accepted Answer

Let's solve the system of equations step by step:

Given:
1) x + y + z = 12   ...(i)
2) x + y – z = 6    ...(ii)
3) x – y + z = 4    ...(iii)

Step 1: Add equations (i) and (ii):
(x + y + z) + (x + y – z) = 12 + 6
=> 2x + 2y = 18
=> x + y = 9   ...(iv)

Step 2: Subtract equation (iii) from equation (i):
(x + y + z) – (x – y + z) = 12 – 4
=> (x + y + z) – x + y – z = 8
=> x + y + z – x + y – z = 8
=> 2y = 8
=> y = 4

But let's double-check this step, as the calculation seems off.

Let’s try a different approach:

From (iv): x + y = 9
So, x = 9 – y

Substitute x in (iii):
(9 – y) – y + z = 4
=> 9 – 2y + z = 4
=> z = 4 + 2y – 9 = 2y – 5

Now substitute x and z in (i):
(9 – y) + y + (2y – 5) = 12
=> 9 + 2y – 5 = 12
=> 2y + 4 = 12
=> 2y = 8
=> y = 4

Now,
x = 9 – y = 9 – 4 = 5
z = 2y – 5 = 2×4 – 5 = 8 – 5 = 3

So, x = 5, y = 4, z = 3

But this matches option 3, which is marked as incorrect. Let's check the equations with these values:

Equation (i): 5 + 4 + 3 = 12 ✔️
Equation (ii): 5 + 4 – 3 = 6 ✔️
Equation (iii): 5 – 4 + 3 = 4 ✔️

So, the correct answer is x = 5, y = 4, z = 3 (Option 3). The initial marking was incorrect.

Final Answer: Option 3 is correct.

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