To find the equation of the line that passes through the points \( (9, -9) \) and \( (10, -5) \), we can follow these steps:
Step 1: Find the slope of the line.
The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points \( (9, -9) \) and \( (10, -5) \):
- \( x_1 = 9, y_1 = -9 \)
- \( x_2 = 10, y_2 = -5 \)
Calculating the slope:
\[ m = \frac{-5 - (-9)}{10 - 9} = \frac{-5 + 9}{1} = \frac{4}{1} = 4 \]
a. Write an equation for the line in point-slope form.
The point-slope form of a line is given by the equation:
\[ y - y_1 = m(x - x_1) \]
Using the point \( (9, -9) \) and the slope \( m = 4 \):
\[ y - (-9) = 4(x - 9) \]
Which simplifies to:
\[ y + 9 = 4(x - 9) \]
b. Rewrite the equation in standard form using integers.
First, we'll expand and rearrange the point-slope form:
\[ y + 9 = 4x - 36 \]
Now, subtract \( y \) and \( 9 \) from both sides:
\[ -4x + y + 9 = -36 \]
This simplifies to:
\[ -4x + y = -36 - 9 \]
Finally, combine the constants:
\[ -4x + y = -45 \]
To get the standard form \( Ax + By = C \) with \( A \) being positive, multiply every term by -1:
\[ 4x - y = 45 \]
Thus, the equation in standard form is:
\[ 4x - y = 45 \]
Summary:
a. Point-slope form: \( y + 9 = 4(x - 9) \)
b. Standard form: \( 4x - y = 45 \)