The graphs of y = –7x + 12 and y = 2/3x –2/3 are shown. which region describes the solution of the system of inequalities y ≤ –7x + 12 and y ≤ –2/3x –2/3 ?

You can draw both lines y=-7x+12 and y=-2/3x –2/3 and define which part of the plane satisfies the unequalities y ≤ –7x + 12 and y ≤ –2/3x –2/3. For example, choosing point (0,0) you can see that
0 ≤ –7·0 + 12 (is true) and 0 ≤ –2/3·0 –2/3 (is false). This means that (0,0) is a solution of the first unequality and isn't a solution of the second unequality, hence (0,0) belongs to the shaded part of the plane defined by the first unequality and doesn't belong to the shaded part defined by the second unequality. Both (black on the image) parts intersect and the common area (red on the picture) is the region that describes the solutions of the system of inequalities.