Question

Which equation of the linear function models the relationship shown in the table. Show your work.

2,21
4,17
5,15
8,9
10,5

(1 pt)
Responses

y=2x+24
y is equal to 2 x plus 24

y=−2x+25
y is equal to negative 2 x plus 25

y=−3x+19
y is equal to negative 3 x plus 19

y=2x+21
y is equal to 2 x plus 21
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Answer 1

To determine the linear function that models the relationship shown in the table, we need to find the slope and y-intercept of the line formed by the data points.

The given points in the table are:
- (2, 21)
- (4, 17)
- (5, 15)
- (8, 9)
- (10, 5)

1. **Calculate the slope (m)**:
The slope (m) of a linear function can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Let's calculate the slope using two points, for example, (2, 21) and (4, 17):
\[
m = \frac{17 - 21}{4 - 2} = \frac{-4}{2} = -2
\]

We can verify this slope using another pair of points, such as (4, 17) and (5, 15):
\[
m = \frac{15 - 17}{5 - 4} = \frac{-2}{1} = -2
\]

This consistent slope confirms that it's correct.

2. **Use the slope to find the equation**:
The general form of a linear equation is:
\[
y = mx + b
\]
where \( m \) is the slope and \( b \) is the y-intercept.

We know:
\[
m = -2
\]
Next, we can find the value of \( b \) by substituting one of the points into the equation. Let's use the point (2, 21):
\[
21 = -2(2) + b
\]
\[
21 = -4 + b
\]
\[
b = 21 + 4 = 25
\]

3. **Write the final equation**:
Now, we can write the complete equation of the line:
\[
y = -2x + 25
\]

The correct equation of the linear function that models the relationship shown in the table is:
\[
\boxed{y = -2x + 25}
\]

This matches one of the response options provided in your original question.