If `x ( x + y + z) = 20 , y ( x + y + z) = 30 and z ( x + y + z) = 50 ` the vaue of 2 (x + y + z) is

Options

A. 20
B. `-10`
C. 15
D. 18

Answer

To solve the problem, we start with the given equations:

1. \( x(x + y + z) = 20 \)
2. \( y(x + y + z) = 30 \)
3. \( z(x + y + z) = 50 \)

Let's denote \( S = x + y + z \). Then we can rewrite the equations as follows:

1. \( xS = 20 \)
2. \( yS = 30 \)
3. \( zS = 50 \)

Now, we can express \( x \), \( y \), and \( z \) in terms of \( S \):

From the first equation:
\[ x = \frac{20}{S} \]

From the second equation:
\[ y = \frac{30}{S} \]

From the third equation:
\[ z = \frac{50}{S} \]

Next, we can substitute these expressions for \( x \), \( y \), and \( z \) back into the equation for \( S \):

\[ S = x + y + z = \frac{20}{S} + \frac{30}{S} + \frac{50}{S} \]

Combining the fractions gives:
\[ S = \frac{20 + 30 + 50}{S} = \frac{100}{S} \]

Now, we can multiply both sides by \( S \) to eliminate the fraction:
\[ S^2 = 100 \]

Taking the square root of both sides, we find:
\[ S = 10 \]

Now, we need to find the value of \( 2(x + y + z) \):
\[ 2S = 2 \times 10 = 20 \]

Thus, the value of \( 2(x + y + z) \) is \( 20 \).