Respuesta :
Question:
Which set of ordered pairs (x,y) could represent a linear function?
[tex]\mathbf{A}= \,\,\left\{(2, 9),\,\,(4, 5),\,\,(6, 1),\,\,(7, -1)\right\}[/tex]
[tex]\mathbf{B}= \,\,\left\{(-1, -8),\,\,(0, -3),\,\,(1, 3),\,\,(2, 8)\right\}[/tex]
[tex]\mathbf{C}= \,\,\left\{(-9, 5),\,\,(-3, 3),\,\,(3, 0),\,\,(9, -2)\right\}[/tex]
[tex]\mathbf{D}= \,\,\left\{(1, -8),\,\,(2, -6),\,\,(3, -3),\,\,(4, 0)\right\}[/tex]
Answer:
[tex]\mathbf{A}= \,\,\left\{(2, 9),\,\,(4, 5),\,\,(6, 1),\,\,(7, -1)\right\}[/tex]
Step-by-step explanation:
Given
Ordered pairs A - D
Required
Which represents a linear function?
To do this, we simply calculate the slope (m) of each ordered pairs.
[tex]m = \frac{y_2 - y_1}{x_2-x_1}[/tex]
Considering A:
[tex]\mathbf{A}= \,\,\left\{(2, 9),\,\,(4, 5),\,\,(6, 1),\,\,(7, -1)\right\}[/tex]
Consider the following pairs
[tex](x_1,y_1) = (2,9)[/tex] and [tex](x_2,y_2) = (4,5)[/tex]
[tex]m = \frac{5 - 9}{4 -2} = \frac{-4}{2} = -2[/tex]
Consider the another pairs
[tex](x_1,y_1) = (2,9)[/tex] and [tex](x_2,y_2) = (6,1)[/tex]
[tex]m = \frac{1 - 9}{6 -2} = \frac{-8}{4} = -2[/tex]
Consider the another pairs
[tex](x_1,y_1) = (2,9)[/tex] and [tex](x_2,y_2) = (7,-1)[/tex]
[tex]m = \frac{-1 - 9}{7 -2} = \frac{-10}{5} = -2[/tex]
The slope is uniform all through.
i.e.
[tex]m = -2[/tex]
Hence, this can be a linear function
Considering B:
[tex]\mathbf{B}= \,\,\left\{(-1, -8),\,\,(0, -3),\,\,(1, 3),\,\,(2, 8)\right\}[/tex]
Consider the following pairs
[tex](x_1,y_1) = (-1,-8)[/tex] and [tex](x_2,y_2) = (0,-3)[/tex]
[tex]m = \frac{-3 - (-8)}{0 - (-1)} = \frac{-3 +8}{0 +1} = \frac{5}{1} = 5[/tex]
Consider the another pairs
[tex](x_1,y_1) = (-1,-8)[/tex] and [tex](x_2,y_2) = (1,3)[/tex]
[tex]m = \frac{3 - (-8)}{1 - (-1)} = \frac{3 +8}{1 +1} = \frac{11}{2} = 5.5[/tex]
The calculated slopes are not equal.
i.e.
[tex]m = 5[/tex] and [tex]m = 5.5[/tex]
Hence, this can't be a linear function
Considering C:
[tex]\mathbf{C}= \,\,\left\{(-9, 5),\,\,(-3, 3),\,\,(3, 0),\,\,(9, -2)\right\}[/tex]
Consider the following pairs
[tex](x_1,y_1) = (-9,5)[/tex] and [tex](x_2,y_2) = (-3,3)[/tex]
[tex]m=\frac{3-5}{-3-(-9)} = \frac{3-5}{-3+9} = \frac{-2}{6} = -\frac{1}{3}[/tex]
Consider the another pairs
[tex](x_1,y_1) = (-9,5)[/tex] and [tex](x_2,y_2) = (3,0)[/tex]
[tex]m=\frac{0-5}{3-(-9)} = \frac{0-5}{3+9} = \frac{-5}{12} = -\frac{5}{12}[/tex]
The calculated slopes are not equal.
i.e.
[tex]m = -\frac{5}{12}[/tex] and [tex]m = -\frac{1}{3}[/tex]
Hence, this can't be a linear function
Considering D:
[tex]\mathbf{D}= \,\,\left\{(1, -8),\,\,(2, -6),\,\,(3, -3),\,\,(4, 0)\right\}[/tex]
Consider the following pairs
[tex](x_1,y_1) = (1,-8)[/tex] and [tex](x_2,y_2) = (2,-6)[/tex]
[tex]m = \frac{-6 - (-8)}{2 - 1} = \frac{6 + 8}{1} = \frac{14}{1} = 14[/tex]
Consider the another pairs
[tex](x_1,y_1) = (1,-8)[/tex] and [tex](x_2,y_2) = (3,-3)[/tex]
[tex]m = \frac{3 - (-8)}{-3 - 1} = \frac{3 + 8}{-4} = \frac{11}{-4} = -\frac{11}{4}[/tex]
The calculated slopes are not equal.
i.e.
[tex]m = -\frac{11}{4}[/tex] and [tex]m = 14[/tex]
Hence, this can't be a linear function
From the calculations above:
Only (A) can be a linear function
