If `x^(2) + 3y^(2) + 4z^(2)+ 19 = 4 sqrt(3) ( x + y + z)`, then the value of `( x - y + 4z)` is:

Options

A. `( 4 sqrt3)/(3)`
B. `( 7 sqrt(3) )/( 2)`
C. `( 10 sqrt(3) )/( 3)`
D. `4 sqrt(3)`

Answer

To solve the equation \( x^2 + 3y^2 + 4z^2 + 19 = 4\sqrt{3}(x + y + z) \) and find the value of \( x - y + 4z \), we can follow these steps:

### Step 1: Rearranging the Equation
Start by rearranging the equation to group all terms on one side:
\[ x^2 + 3y^2 + 4z^2 - 4\sqrt{3}x - 4\sqrt{3}y - 4\sqrt{3}z + 19 = 0 \]