the tables of ordered pairs represent some points on the graphs of two different lines.
Line a
X: -11, -6, -1, 4
Y: 81, 51, 21, -9
Line b
X: -9, -4, 1, 6
Y: 18, 3, -12, -27
Which systems of equations represents the graph of these two lines?
A) x - 6y = -90, x - 3y = 27
B) x + 6y = 90, x + 3y = -27
C) 6x - y = -15, 3x - y = -9
D) 6x + y = 15, 3x + y = -9
To determine which system of equations represents the graph of the two given lines, we can first find the equations of the lines using the points provided.
For Line a:
Using the points (-11, 81) and (4, -9), we can calculate the slope:
m = (y2 - y1) / (x2 - x1)
m = (-9 - 81) / (4 - (-11))
m = -90 / 15
m = -6
The equation of Line a using the point (-11, 81) can be found using the point-slope formula:
y - y1 = m(x - x1)
y - 81 = -6(x + 11)
y - 81 = -6x - 66
y = -6x + 15 (Equation 1)
For Line b:
Using the points (-9, 18) and (6, -27), we can calculate the slope:
m = (y2 - y1) / (x2 - x1)
m = (-27 - 18) / (6 - (-9))
m = -45 / 15
m = -3
The equation of Line b using the point (-9, 18) can be found using the point-slope formula:
y - y1 = m(x - x1)
y - 18 = -3(x + 9)
y - 18 = -3x - 27
y = -3x - 9 (Equation 2)
Now, compare the two equations with the answer choices:
A) x - 6y = -90, x - 3y = 27
Substitute for x and y using Line a equation:
-6(-6x + 15) = -90
36x - 90 = -90
36x = 0
x = 0 (not true)
Substitute for x and y using Line b equation:
-3(-3x - 9) = 27
9x + 27 = 27
9x = 0
x = 0 (not true)
B) x + 6y = 90, x + 3y = -27
Substitute for x and y using Line a equation:
-6 + 15 = 90
9 = 90 (not true)
Substitute for x and y using Line b equation:
-3 - 9 = -27
-12 = -27 (not true)
C) 6x - y = -15, 3x - y = -9
Substitute for x and y using Line a equation:
6(-6) - 21 = -15
-36 - 21 = -15
-57 = -15 (not true)
Substitute for x and y using Line b equation:
3(-3) - 21 = -9
-9 - 18 = -9
-27 = -9 (not true)
D) 6x + y = 15, 3x + y = -9
Substitute for x and y using Line a equation:
6(-6) + 15 = 15
-36 + 15 = 15
-21 = 15 (not true)
Substitute for x and y using Line b equation:
3(-3) + 21 = -9
-9 + 18 = -9
9 = -9 (not true)
None of the answer choices accurately represent the system of equations for the given lines.