The coordinate plane below represents a city. Points A through F are schools in the city.
graph of coordinate plane. Point A is at negative 5, 5. Point B is at negative 4, negative 2. Point C is at 2, 1. Point D is at negative 2, 4. Point E is at 2, 4. Point F is at 3, negative 4.
Part A: Using the graph above, create a system of inequalities that only contain points D and E in the overlapping shaded regions. Explain how the lines will be graphed and shaded on the coordinate grid above. (5 points)
Part B: Explain how to verify that the points D and E are solutions to the system of inequalities created in Part A. (3 points)
Part C: Timothy can only attend a school in his designated zone. Timothy's zone is defined by y < 3x − 3. Explain how you can identify the schools that Timothy is allowed to attend. (2 points)
Part A: To create a system of inequalities that only contains points D and E in the overlapping shaded regions, we need to determine the equations for the lines that separate the regions containing each point.
1. Let's start with point D, which is at (-2, 4). We want to create an equation in the form y < mx + b. To find the equation of the line that passes through point D, we can use the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.
The slope can be found by calculating the rise over the run: (change in y) / (change in x). Using the coordinates (-2, 4) and (0, b), we have:
m = (4 - b) / (-2 - 0) = (4 - b) / -2
Since we have the point (-2, 4) on the line, we can substitute these coordinates into the equation to solve for b:
4 = m(-2) + b
4 = 2m + b
Substituting the expression for m, we get:
4 = 2[(4 - b) / -2] + b
4 = (4 - b) / -1 + b
Simplifying further:
4 = (4 - b) - b
4 = 4 - 2b
2b = 0
b = 0
Therefore, the equation of the line passing through point D is y = (4 - 0) / -2 = -2x.
2. Now let's find the equation for the line that passes through point E, which is at (2, 4). Following the same steps as above:
m = (4 - b) / (2 - 0) = (4 - b) / 2
Substitute the coordinates (2, 4) into the equation:
4 = m(2) + b
4 = 2m + b
Substituting the expression for m, we get:
4 = 2[(4 - b) / 2] + b
4 = (4 - b) + b
Simplifying further:
4 = 4
b can be any value.
Therefore, the equation of the line passing through point E is y = (4 - b) / 2, where b can be any real number.
Now that we have the equations of the lines, we need to determine the inequalities that represent the shaded regions where both points D and E fall into simultaneously.
For point D: Since the line y = -2x passes through point D, all points below this line will satisfy the condition y < -2x.
For point E: Since the line y = (4 - b)/2 does not have a fixed slope, the inequality representing the shaded region that contains point E will be y ≤ (4 - b)/2.
The shaded regions represent the overlapping areas where both points D and E are contained.
Part B: To verify if points D (-2, 4) and E (2, 4) are solutions to the system of inequalities, substitute the coordinates of each point into the inequalities from Part A and check if the inequalities hold true.
For point D: Substitute x = -2 and y = 4 into the inequality y < -2x.
4 < -2(-2)
4 < 4
Since the inequality holds true, point D satisfies the inequality y < -2x.
For point E: Substitute x = 2 and y = 4 into the inequality y ≤ (4 - b)/2.
4 ≤ (4 - b)/2
8 ≤ 4 - b
8 + b ≤ 4
b ≥ -4
Since b can be any real number, including -4, point E satisfies the inequality y ≤ (4 - b)/2.
Therefore, both points D and E are solutions to the system of inequalities.
Part C: To identify the schools Timothy is allowed to attend, we need to substitute the coordinates of each school (A through F) into the inequality y < 3x - 3. If the inequality holds true, Timothy is allowed to attend that particular school.
For example, let's take school A, which is at (-5, 5):
5 < 3(-5) - 3
5 < -15 - 3
5 < -18
Since 5 is not less than -18, school A is not within Timothy's allowed zone.
Repeat this process for each school, substituting the coordinates into the inequality y < 3x - 3. The schools that satisfy this inequality are within Timothy's designated zone, and those are the schools he is allowed to attend.