Which set of ordered pairs left parenthesis, x, comma, y, right parenthesis(x,y) could represent a linear function?
A, equals, left curly bracket, left parenthesis, 0, comma, minus, 8, right parenthesis, comma, left parenthesis, 3, comma, minus, 5, right parenthesis, comma, left parenthesis, 6, comma, minus, 2, right parenthesis, comma, left parenthesis, 9, comma, 1, right parenthesis, right curly bracket
A=
{(0,−8),(3,−5),(6,−2),(9,1)}
B, equals, left curly bracket, left parenthesis, 3, comma, 2, right parenthesis, comma, left parenthesis, 4, comma, 0, right parenthesis, comma, left parenthesis, 7, comma, minus, 6, right parenthesis, comma, left parenthesis, 8, comma, minus, 9, right parenthesis, right curly bracket
B=
{(3,2),(4,0),(7,−6),(8,−9)}
C, equals, left curly bracket, left parenthesis, minus, 9, comma, 7, right parenthesis, comma, left parenthesis, minus, 3, comma, 3, right parenthesis, comma, left parenthesis, 3, comma, minus, 1, right parenthesis, comma, left parenthesis, 9, comma, minus, 4, right parenthesis, right curly bracket
C=
{(−9,7),(−3,3),(3,−1),(9,−4)}
D, equals, left curly bracket, left parenthesis, minus, 4, comma, 4, right parenthesis, comma, left parenthesis, minus, 1, comma, 0, right parenthesis, comma, left parenthesis, 2, comma, minus, 5, right parenthesis, comma, left parenthesis, 5, comma, minus, 9, right parenthesis, right curly bracket
D=
{(−4,4),(−1,0),(2,−5),(5,−9)}

To determine which set of ordered pairs could represent a linear function, we need to check if the slope between each pair of points is consistent.

The slope between any two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as:

\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Let's calculate the slope for each set of ordered pairs:

### Set A: \(\{(0, -8), (3, -5), (6, -2), (9, 1)\}\)
1. Between \((0, -8)\) and \((3, -5)\):
   \[
   \frac{-5 - (-8)}{3 - 0} = \frac{3}{3} = 1
   \]
2. Between \((3, -5)\) and \((6, -2)\):
   \[
   \frac{-2 - (-5)}{6 - 3} = \frac{3}{3} = 1
   \]
3. Between \((6, -2)\) and \((9, 1)\):
   \[
   \frac{1 - (-2)}{9 - 6} = \frac{3}{3} = 1
   \]

All slopes are equal, so **Set A** represents a linear function.

### Set B: \(\{(3, 2), (4, 0), (7, -6), (8, -9)\}\)
1. Between \((3, 2)\) and \((4, 0)\):
   \[
   \frac{0 - 2}{4 - 3} = \frac{-2}{1} = -2
   \]
2. Between \((4, 0)\) and \((7, -6)\):
   \[
   \frac{-6 - 0}{7 - 4} = \frac{-6}{3} = -2
   \]
3. Between \((7, -6)\) and \((8, -9)\):
   \[
   \frac{-9 - (-6)}{8 - 7} = \frac{-3}{1} = -3
   \]

The slope is not consistent, so **Set B** does not represent a linear function.

### Set C: \(\{(-9, 7), (-3, 3), (3, -1), (9, -4)\}\)
1. Between \((-9, 7)\) and \((-3, 3)\):
   \[
   \frac{3 - 7}{-3 - (-9)} = \frac{-4}{6} = -\frac{2}{3}
   \]
2. Between \((-3, 3)\) and \((3, -1)\):
   \[
   \frac{-1 - 3}{3 - (-3)} = \frac{-4}{6} = -\frac{2}{3}
   \]
3. Between \((3, -1)\) and \((9, -4)\):
   \[
   \frac{-4 - (-1)}{9 - 3} = \frac{-3}{6} = -\frac{1}{2}
   \]

The slope is not consistent, so **Set C** does not represent a linear function.

### Set D: \(\{(-4, 4), (-1, 0), (2, -5), (5, -9)\}\)
1. Between \((-4, 4)\) and \((-1, 0)\):
   \[
   \frac{0 - 4}{-1 - (-4)} = \frac{-4}{3} = -\frac{4}{3}
   \]
2. Between \((-1, 0)\) and \((2, -5)\):
   \[
   \frac{-5 - 0}{2 - (-1)} = \frac{-5}{3} = -\frac{5}{3}
   \]
3. Between \((2, -5)\) and \((5, -9)\):
   \[
   \frac{-9 - (-5)}{5 - 2} = \frac{-4}{3} = -\frac{4}{3}
   \]

The slope is not consistent, so **Set D** does not represent a linear function.

### Conclusion:
Only **Set A** represents a linear function because the slope between all pairs of points is consistent. 

Would you like more details or have any other questions?

Here are five related questions to consider:
1. How can you identify a non-linear function from a set of points?
2. What is the equation of the line represented by Set A?
3. How do you calculate the y-intercept of a line from its slope and a point?
4. Can a function be linear if its slope is zero?
5. How do parallel lines relate to their slopes?

**Tip:** When verifying linearity in a function, check if the difference in y-values over the difference in x-values is constant across all pairs of points.

Learn how to identify linear functions from sets of ordered pairs with a detailed explanation. Analyze sets A, B, C, and D to understand the consistency of slopes and determine which sets represent linear functions. Suitable for Grades 7-9.

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