To determine the equation of a line that is parallel to a given line and passes through a specific point, we need to follow these steps:
### Step 1: Identify the given line and its equation
The given line has the equation:
[tex]\[ 3x - 4y = -17 \][/tex]
### Step 2: Determine the slope of the given line
Parallel lines have the same slope. However, in this case, we'll use the general form of the line equation directly since the coefficients will be the same for parallel lines.
### Step 3: Use the general form of the equation for parallel lines
Since parallel lines have identical coefficients for [tex]\( x \)[/tex] and [tex]\( y \)[/tex], the general form of the equation for a line parallel to [tex]\( 3x - 4y = -17 \)[/tex] will be:
[tex]\[ 3x - 4y = C \][/tex]
where [tex]\( C \)[/tex] is a constant that we'll determine next.
### Step 4: Substitute the given point into the equation to find [tex]\( C \)[/tex]
We are given the point [tex]\( (-3, 2) \)[/tex]. Substitute [tex]\( x = -3 \)[/tex] and [tex]\( y = 2 \)[/tex] into the equation [tex]\( 3x - 4y = C \)[/tex]:
[tex]\[ 3(-3) - 4(2) = C \][/tex]
[tex]\[ -9 - 8 = C \][/tex]
[tex]\[ -17 = C \][/tex]
### Step 5: Write the final equation
Now that we have found [tex]\( C = -17 \)[/tex], the equation of the line parallel to [tex]\( 3x - 4y = -17 \)[/tex] and passing through the point [tex]\( (-3, 2) \)[/tex] is:
[tex]\[ 3x - 4y = -17 \][/tex]
Thus, the equation of the desired line is:
[tex]\[ 3x - 4y = -17 \][/tex]
Given the options:
1. [tex]\( 3x - 4y = -17 \)[/tex]
2. [tex]\( 3x - 4y = -20 \)[/tex]
3. [tex]\( 4x + 3y = -2 \)[/tex]
4. [tex]\( 4x + 3y = -6 \)[/tex]
The correct answer is:
[tex]\[ \boxed{3x - 4y = -17} \][/tex]
