Answer: $263$

Explanation:1. **Understand the problem:** We have a system of two linear equations. We need to find the $x$-coordinate of the solution and then determine the value of $p$ based on the given form $x = \dfrac{p}{59}$.

2. **Choose a method to solve the system for $x$:** We can use the elimination method. To eliminate $y$, we need to make the coefficients of $y$ opposites. The least common multiple of $21$ and $14$ is $42$. Multiply the first equation by $2$: $2 \times (22x-21y=43) \Rightarrow 44x - 42y = 86$ (Equation 3) Multiply the second equation by $3$: $3 \times (5x+14y=59) \Rightarrow 15x + 42y = 177$ (Equation 4)

3. **Eliminate $y$ by adding the modified equations:** Add Equation (3) and Equation (4): $(44x - 42y) + (15x + 42y) = 86 + 177$ $44x + 15x - 42y + 42y = 263$ $59x = 263$

4. **Solve for $x$:** $x = \dfrac{263}{59}$

5. **Determine the value of $p$:** We are given that the $x$-coordinate of the solution is $x = \dfrac{p}{59}$. We found $x = \dfrac{263}{59}$. Comparing these two expressions for $x$, we have: $\dfrac{p}{59} = \dfrac{263}{59}$ Therefore, $p = 263$.

6. **Conclusion:** The value of $p$ is $263$.