Introduction:
In this problem, we are given a system of equations and we need to verify graphically whether the system has a solution or not. The given system of equations is:
Equation 1: 2x + 3y = 10
Equation 2: 4x + 6y = 12

Plotting the equations:
To verify graphically, we will plot the two equations on a graph and see if they intersect at a common point, which represents a solution, or if they are parallel lines, which indicates no solution.

Step 1: Find the intercepts:
To plot the equations, we first find the intercepts of each equation. The intercepts are the points where the lines cross the x-axis and y-axis.

Equation 1:
To find the x-intercept, we set y = 0 in Equation 1 and solve for x:
2x + 3(0) = 10
2x = 10
x = 5

So, the x-intercept for Equation 1 is (5, 0).

To find the y-intercept, we set x = 0 in Equation 1 and solve for y:
2(0) + 3y = 10
3y = 10
y = 10/3

So, the y-intercept for Equation 1 is (0, 10/3).

Equation 2:
To find the x-intercept, we set y = 0 in Equation 2 and solve for x:
4x + 6(0) = 12
4x = 12
x = 3

So, the x-intercept for Equation 2 is (3, 0).

To find the y-intercept, we set x = 0 in Equation 2 and solve for y:
4(0) + 6y = 12
6y = 12
y = 2

So, the y-intercept for Equation 2 is (0, 2).

Step 2: Plot the points:
Now that we have the intercepts for both equations, we can plot these points on a graph.

Equation 1:
Plot the x-intercept (5, 0) and the y-intercept (0, 10/3).

Equation 2:
Plot the x-intercept (3, 0) and the y-intercept (0, 2).

Step 3: Draw the lines:
Connect the two intercept points for each equation to obtain the lines.

Analysis:
After plotting the lines, we observe that the lines are parallel and do not intersect at any point. This means that there is no solution to the given system of equations.

Conclusion:
Therefore, graphically verifying the system of equations 2x + 3y = 10 and 4x + 6y = 12 confirms that the system is inconsistent and has no solution. The lines representing the equations are parallel and never intersect.