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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

(10,7)
(10,7)

(15,7)
(15,7)

(5,6)
(5,6)

(0,-3)
(0,-3)

(0,5)
(0,5)

(-3,-12)
(-3,-12)

(3,3)
(3,3)

(11,12)
(11,12)

1 answer

To determine which coordinate points are solutions to the system of inequalities:

  1. Inequality 1: \( y < 2x - 3 \)
  2. Inequality 2: \( x - 3y < -6 \)

We will substitute each point (x, y) into both inequalities and check if they satisfy both.

Testing Each Point:

Point (10, 7):

  • For \( y < 2x - 3 \): \( 7 < 2(10) - 3 \) → \( 7 < 20 - 3 \) → \( 7 < 17 \) (True)
  • For \( x - 3y < -6 \): \( 10 - 3(7) < -6 \) → \( 10 - 21 < -6 \) → \( -11 < -6 \) (False)

Point (15, 7):

  • For \( y < 2x - 3 \): \( 7 < 2(15) - 3 \) → \( 7 < 30 - 3 \) → \( 7 < 27 \) (True)
  • For \( x - 3y < -6 \): \( 15 - 3(7) < -6 \) → \( 15 - 21 < -6 \) → \( -6 < -6 \) (False)

Point (5, 6):

  • For \( y < 2x - 3 \): \( 6 < 2(5) - 3 \) → \( 6 < 10 - 3 \) → \( 6 < 7 \) (True)
  • For \( x - 3y < -6 \): \( 5 - 3(6) < -6 \) → \( 5 - 18 < -6 \) → \( -13 < -6 \) (True)

Point (0, -3):

  • For \( y < 2x - 3 \): \( -3 < 2(0) - 3 \) → \( -3 < -3 \) (False)
  • For \( x - 3y < -6 \): \( 0 - 3(-3) < -6 \) → \( 0 + 9 < -6 \) → \( 9 < -6 \) (False)

Point (0, 5):

  • For \( y < 2x - 3 \): \( 5 < 2(0) - 3 \) → \( 5 < -3 \) (False)
  • For \( x - 3y < -6 \): \( 0 - 3(5) < -6 \) → \( 0 - 15 < -6 \) → \( -15 < -6 \) (True)

Point (-3, -12):

  • For \( y < 2x - 3 \): \( -12 < 2(-3) - 3 \) → \( -12 < -6 - 3 \) → \( -12 < -9 \) (True)
  • For \( x - 3y < -6 \): \( -3 - 3(-12) < -6 \) → \( -3 + 36 < -6 \) → \( 33 < -6 \) (False)

Point (3, 3):

  • For \( y < 2x - 3 \): \( 3 < 2(3) - 3 \) → \( 3 < 6 - 3 \) → \( 3 < 3 \) (False)
  • For \( x - 3y < -6 \): \( 3 - 3(3) < -6 \) → \( 3 - 9 < -6 \) → \( -6 < -6 \) (False)

Point (11, 12):

  • For \( y < 2x - 3 \): \( 12 < 2(11) - 3 \) → \( 12 < 22 - 3 \) → \( 12 < 19 \) (True)
  • For \( x - 3y < -6 \): \( 11 - 3(12) < -6 \) → \( 11 - 36 < -6 \) → \( -25 < -6 \) (True)

Summary of Results

The points that satisfy both inequalities are:

  • (5, 6)
  • (11, 12)

Therefore, the solutions to the system of linear inequalities are:

  • (5, 6)
  • (11, 12)