Question

Which equation of the linear function models the relationship shown in the table below. Show your work.
X 2, 4, 5, 8, 10. Y 21, 17, 15, 9, 5.

Answer 1

To determine the equation of the linear function that models the relationship between \( X \) and \( Y \) given in the table, we need to find the slope (m) and the y-intercept (b) of the linear equation in the form \( Y = mX + b \).

### Step 1: Determine the slope (m)
The slope (m) of a line is calculated using the formula:

\[
m = \frac{Y_2 - Y_1}{X_2 - X_1}
\]

We can pick two points from the table. Let's choose the points \( (2, 21) \) and \( (10, 5) \):

- \( Y_1 = 21 \)
- \( Y_2 = 5 \)
- \( X_1 = 2 \)
- \( X_2 = 10 \)

Now, plug in these values:

\[
m = \frac{5 - 21}{10 - 2} = \frac{-16}{8} = -2
\]

### Step 2: Use one of the points and the slope to find the y-intercept (b)
We can use the point \( (2, 21) \) and the slope \( m = -2 \) to solve for \( b \).

The equation of the line in point-slope form is:

\[
Y - Y_1 = m(X - X_1)
\]

Plugging in the values we have:

\[
Y - 21 = -2(X - 2)
\]

### Step 3: Simplify to find the equation in slope-intercept form
Distributing the \(-2\):

\[
Y - 21 = -2X + 4
\]

Now, add 21 to both sides:

\[
Y = -2X + 25
\]

### Conclusion
The equation of the linear function that models the relationship between \( X \) and \( Y \) is:

\[
Y = -2X + 25
\]