We are given a triangle $ABC$ and a point $S$ which is the circumcenter. $CP = 7x - 1$ and $PB = 6x + 3$. We need to find the value of $x$ and the length of $CP$.

1. Problem Description

We are given a triangle $ABC$ and a point $S$ which is the circumcenter. $CP = 7x - 1$ and $PB = 6x + 3$. We need to find the value of $x$ and the length of $CP$.

2. Solution Steps

Since $S$ is the circumcenter of triangle $ABC$, the point $P$ must be the midpoint of the side $CB$.

Therefore, $CP = PB$.

We are given $CP = 7x - 1$ and $PB = 6x + 3$.

So, we can set up the equation:

$7x - 1 = 6x + 3$

Subtracting $6x$ from both sides, we get:

$7x - 6x - 1 = 6x - 6x + 3$

$x - 1 = 3$

Adding $1$ to both sides, we get:

$x - 1 + 1 = 3 + 1$

$x = 4$

Now, we can find the length of $CP$ by substituting $x = 4$ into the expression for $CP$:

$CP = 7x - 1$

$CP = 7(4) - 1$

$CP = 28 - 1$

$CP = 27$

3. Final Answer

$x = 4$

$CP = 27$