Question

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.


y – 9 = 4(x – 9); –4x + y = 45


y + 9 = 4(x + 9); –4x + y = –45


y – 9 = 4(x + 9); –4x + y = 45


y + 9 = 4(x – 9); –4x + y = –45

Answer 1

To write the equation of a line that passes through the points (9, -9) and (10, -5), we will follow the steps mentioned in your query.

### 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 given points (9, -9) and (10, -5):

\[
m = \frac{-5 - (-9)}{10 - 9} = \frac{-5 + 9}{1} = \frac{4}{1} = 4
\]

### Step 2: Write the equation in point-slope form
Point-slope form of the equation of a line is given by:

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

Using the point (9, -9) and the slope \(m = 4\):

\[
y - (-9) = 4(x - 9)
\]
\[
y + 9 = 4(x - 9)
\]

### Step 3: Rewrite the equation in standard form
Now we will convert the point-slope equation \(y + 9 = 4(x - 9)\) into standard form \(Ax + By = C\).

Starting from the point-slope form:

1. Distribute on the right side:
\[
y + 9 = 4x - 36
\]

2. Move all terms to one side:
\[
-4x + y + 9 + 36 = 0
\]
\[
-4x + y + 45 = 0
\]

3. Rearranging gives:
\[
-4x + y = -45
\]

To write it in a more standard form, we can multiply through by -1:

\[
4x - y = 45
\]

Thus, the standard form is:
\[
-4x + y = -45 \quad (in\ standard\ form\ with\ positive\ A)
\]

### Summary of Answers:
a. The point-slope form of the equation is:
\[
y + 9 = 4(x - 9)
\]

b. The standard form of the equation is:
\[
-4x + y = -45
\]

### Final check against provided options:
From the options given, your equation in point-slope form is:
\[
y + 9 = 4(x - 9) \quad \text{and in standard form:} -4x + y = -45
\]
These correspond to the option:
**y + 9 = 4(x – 9); –4x + y = –45**