Question: Solve the following system of linear equations graphically:

2x + 3y = 12
4x - 3y = 10

Solution:
To solve the given system of linear equations graphically, we need to plot the graphs of each equation on the Cartesian plane and find their point of intersection.

Step 1: Rewrite the given system of equations in the slope-intercept form (y = mx + c), where m is the slope and c is the y-intercept.

Given equations:
2x + 3y = 12 ----(1)
4x - 3y = 10 ----(2)

Rewriting equation (1) in slope-intercept form:
3y = -2x + 12
y = (-2/3)x + 4 ----(3)

Rewriting equation (2) in slope-intercept form:
-3y = -4x + 10
y = (4/3)x - 10/3 ----(4)

Step 2: Plotting the graphs of equations (3) and (4) on the Cartesian plane.

To plot the graphs, we can start by choosing suitable values of x and finding the corresponding values of y.

For equation (3):
When x = 0,
y = (-2/3)(0) + 4 = 4 (point A)
When x = 3,
y = (-2/3)(3) + 4 = 2 (point B)

For equation (4):
When x = 0,
y = (4/3)(0) - 10/3 = -10/3 (point C)
When x = 3,
y = (4/3)(3) - 10/3 = 2 (point D)

Now, plot points A, B, C, and D on the Cartesian plane.

Step 3: Drawing the lines passing through the plotted points.

Draw a line passing through points A and B for equation (3), and draw another line passing through points C and D for equation (4).

Step 4: Finding the point of intersection.

The point where the two lines intersect is the solution to the given system of equations.

Step 5: Reading the coordinates of the point of intersection.

From the graph, we can see that the lines intersect at the point (3, 2).

Step 6: Verifying the solution.

Substitute the values of x and y from the point of intersection into the given equations and check if both equations are satisfied.

For equation (1):
2(3) + 3(2) = 6 + 6 = 12 (satisfied)

For equation (2):
4(3) - 3(2) = 12 - 6 = 6 (satisfied)

Therefore, the point of intersection (3, 2) is the solution to the given system of linear equations.

Summary:
The given system of linear equations is solved graphically by plotting the equations on the Cartesian plane and finding the point of intersection. The solution is verified by substituting the coordinates of the point of intersection into the original equations.