To solve the system of equations by graphing, we need to handle the given equations:

1. $2x - y = 5$
2. $x + 2y = 10$

Steps:

1. Rewrite each equation in slope-intercept form ($y = mx + b$):

   • For $2x - y = 5$: $-y = -2x + 5 \implies y = 2x - 5$

   • For $x + 2y = 10$: $2y = -x + 10 \implies y = -\frac{1}{2}x + 5$

2. Graph these equations:

   • $y = 2x - 5$ has:

     • Slope $m = 2$
     • Y-intercept $b = -5$

   • $y = -\frac{1}{2}x + 5$ has:

     • Slope $m = -\frac{1}{2}$
     • Y-intercept $b = 5$

3. Find the point of intersection:

   • From the equations, set $2x - 5 = -\frac{1}{2}x + 5$.
   • Combine terms: $2x + \frac{1}{2}x = 10 \implies \frac{5}{2}x = 10 \implies x = 4$
   • Substitute $x = 4$ into one of the equations, e.g., $y = 2x - 5$: $y = 2(4) - 5 = 8 - 5 = 3$

   So, the solution is $(4, 3)$.

4. Answer:

   • The solution is $\mathbf{(4, 3)}$.
   • This corresponds to option $A$.

Would you like more details or help understanding any of these steps?

Related Questions:

1. What are the steps to rewrite equations in slope-intercept form?
2. How do you identify the slope and y-intercept of a linear equation?
3. What happens when two lines have the same slope but different intercepts?
4. How do you verify a solution to a system of equations graphically and algebraically?
5. Can a system of equations have no solution or infinitely many solutions? How can you tell?

Tip:

When graphing, always double-check the slope and intercept values to ensure accuracy in plotting.