`3x =x+14`
`sqrt(3z ^(2) -11)+2x =22`
If `z gt 0,` what is the value of z ?

Options

A. 1
B. 3
C. 5
D. 8

Answer

To solve the equations given in the question, we will follow these steps:

### Step 1: Solve for x in the first equation
We start with the equation:
\[ 3x = x + 14 \]

To isolate \( x \), we can subtract \( x \) from both sides:
\[ 3x - x = 14 \]
This simplifies to:
\[ 2x = 14 \]

Now, divide both sides by 2:
\[ x = \frac{14}{2} = 7 \]

### Step 2: Substitute x into the second equation
Now that we have the value of \( x \), we substitute it into the second equation:
\[ \sqrt{3z^2 - 11} + 2x = 22 \]

Substituting \( x = 7 \):
\[ \sqrt{3z^2 - 11} + 2(7) = 22 \]
This simplifies to:
\[ \sqrt{3z^2 - 11} + 14 = 22 \]

### Step 3: Isolate the square root
Next, we isolate the square root by subtracting 14 from both sides:
\[ \sqrt{3z^2 - 11} = 22 - 14 \]
This simplifies to:
\[ \sqrt{3z^2 - 11} = 8 \]

### Step 4: Square both sides
To eliminate the square root, we square both sides:
\[ (\sqrt{3z^2 - 11})^2 = 8^2 \]
This gives us:
\[ 3z^2 - 11 = 64 \]

### Step 5: Solve for z^2
Now, we can solve for \( z^2 \) by first adding 11 to both sides:
\[ 3z^2 = 64 + 11 \]
This simplifies to:
\[ 3z^2 = 75 \]

Next, divide both sides by 3:
\[ z^2 = \frac{75}{3} = 25 \]

### Step 6: Take the square root
Now we take the square root of both sides:
\[ z = \pm \sqrt{25} \]
This results in:
\[ z = \pm 5 \]

### Step 7: Consider the condition z > 0
Since we are given that \( z > 0 \), we only take the positive solution:
\[ z = 5 \]

### Final Answer
Thus, the value of \( z \) is:
\[ z = 5 \]

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