Question

8.
A line passes through (9, –9) and (10, –5).
a. Write an equation for the line in point-slope form.
b. Rewrite the equation in standard form using integers.

Answer 1

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: \( 4x - y = 45 \)