Identify which coordinate points (x,y) below are solutions to the following system of linear inequalities (Select ALL that apply): y<2x-3 x-3y<-6 (3 points) Responses

Question

(0,-3) (0,-3)
(10,7) (10,7)
(3,3) (3,3)
(11,12) (11,12)
(-3,-12) (-3,-12)
(0,5) (0,5)
(15,7) (15,7)
(5,6)

Answer 1

To determine which coordinate points are solutions to the system of inequalities \(y < 2x - 3\) and \(x - 3y < -6\), we can evaluate each point against both inequalities.

1. Evaluating the inequality \(y < 2x - 3\):

• Rearranging gives us \(y + 3 < 2x\) or \(y < 2x - 3\).
• Points that satisfy this inequality will fall below the line \(y = 2x - 3\).

2. Evaluating the inequality \(x - 3y < -6\):

• Rearranging gives us \(x + 6 < 3y\) or \(y > \frac{x + 6}{3}\).
• Points that satisfy this inequality will fall above the line \(y = \frac{x+6}{3}\).

Now let’s evaluate each point:

1. (0, -3):

   • For \(y < 2x - 3\): \(-3 < 2(0) - 3 \Rightarrow -3 < -3\) (False).
   • For \(x - 3y < -6\): \(0 - 3(-3) < -6 \Rightarrow 0 + 9 < -6\) (False).
   • Not a solution.

2. (10, 7):

   • For \(y < 2x - 3\): \(7 < 2(10) - 3 \Rightarrow 7 < 20 - 3 \Rightarrow 7 < 17\) (True).
   • For \(x - 3y < -6\): \(10 - 3(7) < -6 \Rightarrow 10 - 21 < -6 \Rightarrow -11 < -6\) (False).
   • Not a solution.

3. (3, 3):

   • For \(y < 2x - 3\): \(3 < 2(3) - 3 \Rightarrow 3 < 6 - 3 \Rightarrow 3 < 3\) (False).
   • For \(x - 3y < -6\): \(3 - 3(3) < -6 \Rightarrow 3 - 9 < -6 \Rightarrow -6 < -6\) (False).
   • Not a solution.

4. (11, 12):

   • For \(y < 2x - 3\): \(12 < 2(11) - 3 \Rightarrow 12 < 22 - 3 \Rightarrow 12 < 19\) (True).
   • For \(x - 3y < -6\): \(11 - 3(12) < -6 \Rightarrow 11 - 36 < -6 \Rightarrow -25 < -6\) (True).
   • This is a solution.

5. (-3, -12):

   • For \(y < 2x - 3\): \(-12 < 2(-3) - 3 \Rightarrow -12 < -6 - 3 \Rightarrow -12 < -9\) (False).
   • For \(x - 3y < -6\): \(-3 - 3(-12) < -6 \Rightarrow -3 + 36 < -6 \Rightarrow 33 < -6\) (False).
   • Not a solution.

6. (0, 5):

   • For \(y < 2x - 3\): \(5 < 2(0) - 3 \Rightarrow 5 < -3\) (False).
   • For \(x - 3y < -6\): \(0 - 3(5) < -6 \Rightarrow 0 - 15 < -6 \Rightarrow -15 < -6\) (True).
   • Not a solution.

7. (15, 7):

   • For \(y < 2x - 3\): \(7 < 2(15) - 3 \Rightarrow 7 < 30 - 3 \Rightarrow 7 < 27\) (True).
   • For \(x - 3y < -6\): \(15 - 3(7) < -6 \Rightarrow 15 - 21 < -6 \Rightarrow -6 < -6\) (False).
   • Not a solution.

8. (5, 6):

   • For \(y < 2x - 3\): \(6 < 2(5) - 3 \Rightarrow 6 < 10 - 3 \Rightarrow 6 < 7\) (True).
   • For \(x - 3y < -6\): \(5 - 3(6) < -6 \Rightarrow 5 - 18 < -6 \Rightarrow -13 < -6\) (True).
   • This is a solution.

Final Solutions:

The points that are solutions to the system of inequalities are:

• (11, 12)
• (5, 6)