If `x + y + z = 15, xy + yz + zx = 75,` then `(x + 4y + z)/(3z)` is equal to :

Options

A. A)1
B. B)0
C. C)2
D. D)`-1`

Answer

To solve the problem, we need to find the value of \((x + 4y + z)/(3z)\) given the equations:

1. \(x + y + z = 15\)
2. \(xy + yz + zx = 75\)

### Step 1: Express \(x\), \(y\), and \(z\) in terms of a common variable

Let's assume \(x = 5k\), \(y = 5k\), and \(z = 5k\) for some \(k\). This assumption is made to simplify our calculations.

### Step 2: Substitute into the first equation

Substituting into the first equation:

\[ x + y + z = 5k + 5k + 5k = 15k \]

Setting this equal to 15 gives:

\[ 15k = 15 \implies k = 1 \]

Thus, we have:

\[ x = 5(1) = 5, \quad y = 5(1) = 5, \quad z = 5(1) = 5 \]

### Step 3: Verify the second equation

Now, we check the second equation:

\[ xy + yz + zx = 5 \cdot 5 + 5 \cdot 5 + 5 \cdot 5 = 25 + 25 + 25 = 75 \]

This confirms that our values satisfy both equations.

### Step 4: Calculate \((x + 4y + z)/(3z)\)

Now we substitute \(x\), \(y\), and \(z\) into the expression we need to evaluate:

\[ \frac{x + 4y + z}{3z} = \frac{5 + 4(5) + 5}{3(5)} \]

Calculating the numerator:

\[ 5 + 20 + 5 = 30 \]

Calculating the denominator:

\[ 3(5) = 15 \]

Now we can compute the final value:

\[ \frac{30}{15} = 2 \]

### Final Answer

Thus, the value of \(\frac{x + 4y + z}{3z}\) is \(2\).

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