Question

*******ASAP******

The coordinate plane below represents a city. Points A through F are schools in the city.
graph of coordinate plane. Point A is at 1, 3. Point B is at 3, 1. Point C is at 3, negative 3. Point D is at negative 4, 2. Point E is at negative 1, 5. Point F is at negative 3, negative 3.

Part A: Using the graph above, create a system of inequalities that only contains points B and C in the overlapping shaded regions. Explain how the lines will be graphed and shaded on the coordinate grid above.

Part B: Explain how to verify that the points B and C are solutions to the system of inequalities created in Part A.

Part C: Lisa can only attend a school in her designated zone. Lisa's zone is defined by y > 2x + 5. Explain how you can identify the schools that Lisa is allowed to attend.

Answer 1

Part A: We have the points A = ( 1,3 ) , B = ( 3,1 ) and C = ( 3,-3 ).

We will first find the equation of line AB and AC.

Now, the slope of AB is
`(1-3)/(3-1) =(-2)/(2) =-1`.

So, substituting in y = mx + b along with the point ( 1,3 ) gives,

3 = -1 + b i.e. b = 4.

So, the equation of AB is y = -x + 4.

Further, slope of AC is
`(-3-3)/(3-1) =(-6)/(2) =-3`.

Again, substituting in y = mx + b along with the point ( 1,3 ) gives,

3 = -3 + b i.e. b = 6

So, the equation of AC is y = -3x + 6.

Using 'zero test', we get that,

y = -x + 4 gives 0 = 4, which is not true.

y = -3x + 6 gives 0 = 6, which is not true.

Hence, the solution region will be away from the origin as shown in the figure 1 below.

Thus, we get the system y>-x+4 and y>-3x+6 containing point B and C.

Part B: We have the equations y = -x + 4 and y = -3x + 6.

Substitute the points B = ( 3,1 ) and C = ( 3,-3 ) into these equations respectively, we get,

y = -x + 4 gives y = -3 + 4 i.e. y = 1

y = -3x + 6 gives y = -3 × 3 + 6 i.e. y = -9 + 6 i.e. y = -3.

Hence, points B and C are the solutions of the system in Part A.

Part C: After plotting y>2x+5, we ca see from the second figure that the points included in the shaded region are D = ( -4,2 ) and E = (-1, 5).

So, Lisa is allowed to attend the schools D and E.