Question

Suppose that the water level of a river is 45 feet after 3 days of no rain and 10 feet after 10 days of no rain. Given the two points we know: (3, 45) and (10, 10), what would the point slope equation look like?
A. (y - 10) = -5(x + 10)
B. (y - 3) = -5(x - 45)
C. (y + 10) = -5(x - 10)
D. (y - 45) = -5(x - 3)

Answer 1

To find the point-slope equation of the line, we first need to determine the slope using the two points \((3, 45)\) and \((10, 10)\).

The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

Substituting in our points, we have:

\[
y_1 = 45, \quad x_1 = 3, \quad y_2 = 10, \quad x_2 = 10
\]

Now, calculating the slope:

\[
m = \frac{10 - 45}{10 - 3} = \frac{-35}{7} = -5
\]

Now that we have the slope \(m = -5\), we can write the point-slope form of the equation using one of the points. Let's use the point \((3, 45)\).

The point-slope form is given by:

\[
y - y_1 = m(x - x_1)
\]

Substituting our values:

\[
y - 45 = -5(x - 3)
\]

Looking at the choices provided:

A. \((y - 10) = -5(x + 10)\)

B. \((y - 3) = -5(x - 45)\)

C. \((y + 10) = -5(x - 10)\)

D. \((y - 45) = -5(x - 3)\)

The correct answer is:

**D. \((y - 45) = -5(x - 3)\)**