Question

Solve the system of inequalities by graphing.
[1]: y≤–7x+3
[2]: y≤–1/6x+7
Select a line to change it between solid and dotted. Select a region to shade it.

Answer 1

Solution by graphing (instructions you can use in the graphing tool)

1. Draw both boundary lines as solid (because both inequalities are "≤"):
- Line 1: y = −7x + 3 (y-intercept (0,3), x-intercept (3/7)).
- Line 2: y = −(1/6)x + 7 (y-intercept (0,7), x-intercept 42).

2. Mark their intersection point by solving
−7x + 3 = −(1/6)x + 7 → multiply by 6: −42x + 18 = −x + 42 → −41x = 24 → x = −24/41.
Then y = −7(−24/41) + 3 = 291/41.
Intersection: (−24/41, 291/41) ≈ (−0.585, 7.098).

3. Choose a test point to decide which side of each line to shade. Use (0,0):
0 ≤ −7(0) + 3 (true) and 0 ≤ −(1/6)(0) + 7 (true), so the region containing (0,0) satisfies both inequalities.

4. Shade the region that is below both lines (the overlap of the two half-planes). Equivalently, the solution set is
{(x,y) | y ≤ min(−7x + 3, −(1/6)x + 7)}.

So: both lines solid, and shade the region below them (the intersection region that includes the origin).