Let's solve this step by step.
Step 1: Find the slope of the line
To find the slope of the line that passes through the points (9, -9) and (10, -5), we use the slope formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting in the coordinates:
\[ m = \frac{-5 - (-9)}{10 - 9} = \frac{-5 + 9}{1} = \frac{4}{1} = 4 \]
Step 2: Write the equation in point-slope form
The point-slope form of a line is given by:
\[ y - y_1 = m(x - x_1) \]
Using the point (9, -9) and the slope \( m = 4 \):
\[ y - (-9) = 4(x - 9) \]
Which simplifies to:
\[ y + 9 = 4(x - 9) \]
So, the equation in point-slope form is:
\[ y + 9 = 4(x - 9) \]
Step 3: Rewrite the equation in standard form
The standard form of a linear equation is \( Ax + By = C \), where A, B, and C are integers.
Starting from our point-slope equation:
\[ y + 9 = 4(x - 9) \]
Distributing the 4:
\[ y + 9 = 4x - 36 \]
Next, we will subtract \(4x\) from both sides:
\[ -4x + y + 9 = -36 \]
Now, we will subtract 9 from both sides:
\[ -4x + y = -36 - 9 \] \[ -4x + y = -45 \]
To make the coefficient of \(x\) positive, we can multiply the entire equation by -1:
\[ 4x - y = 45 \]
So, the standard form of the equation is:
\[ -4x + y = -45 \]
Review the Options
The correct answer options are:
- \(y + 9 = 4(x - 9)\); \(-4x + y = -45\)
This matches our found point-slope and standard forms.
So the final answer is:
Option: \(y + 9 = 4(x - 9); -4x + y = -45\).