Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region.

Solution:

Given equations are x – y + 1 = 0 and 3x + 2y – 12 = 0.

Let’s consider the equation x – y + 1 = 0.

y = x + 1

x	0	1	2
y	1	2	3

Now, consider the second equation 3x + 2y – 12 = 0

2y = 12 – 3x

y = (12 – 3x)/2

x	0	2	4
y	6	3	0

Hence, the graphical representation of these equations is as follows.

From the figure, it can be seen that these lines are intersecting each other at points (2, 3) and the x-axis at (−1, 0) and (4, 0).

Therefore, the vertices of the triangle are (2, 3), (−1, 0), and (4, 0).