$S$ is the circumcenter of triangle $ABC$. $CP = 7x - 1$ and $PB = 6x + 3$. Find the value of $x$ and $CP$. Since $S$ is the circumcenter, $P$ is the midpoint of $CB$. Thus, $CP = PB$.

1. Problem Description

$S$ is the circumcenter of triangle $ABC$. $CP = 7x - 1$ and $PB = 6x + 3$. Find the value of $x$ and $CP$. Since $S$ is the circumcenter, $P$ is the midpoint of $CB$. Thus, $CP = PB$.

2. Solution Steps

Since $S$ is the circumcenter of triangle $ABC$, $P$ is the midpoint of $CB$. Therefore, $CP = PB$.

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

Setting $CP = PB$, we have:

$7x - 1 = 6x + 3$

Subtract $6x$ from both sides:

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

$x - 1 = 3$

Add $1$ to both sides:

$x - 1 + 1 = 3 + 1$

$x = 4$

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

$CP = 7x - 1$

$CP = 7(4) - 1$

$CP = 28 - 1$

$CP = 27$

3. Final Answer

$x = 4$

$CP = 27$