Answer:
Point-slope equation: y - 5 = 7/2(x + 9)
General form equation: -7x + 2y - 73 = 0
Explanation:
Relationship between the slopes of parallel lines:
- We can determine whether two lines are parallel if their slopes are the same.
- This means we'll first need to find the slope of 7x - 2y - 3 = 0.
Identifying the form of 7x - 2y - 3 = 0 and the general equation of the slope-intercept form:
7x - 2y - 3 = 0 is in the general form of a line, whose general equation is given by:
Ax + By + C = 0, where
- A, B, and C are constants.
The simplest way to find the slope of a line in general form is to convert it to slope-intercept form, whose general equation is given by:
y = mx + b, where
- (x, y) are any point on the line,
- m is the slope,
- and b is the y-intercept.
Converting 7x - 2y - 3 = 0 to slope-intercept form:
Now we can convert 7x - 2y - 3 = 0 to slope-intercept form and determine its slope by isolating y:
(7x - 2y - 3 = 0) + 3 - 7x
(-2y = -7x + 3) / -2
y = 7/2x - 3/2
Thus, 7/2 is the slope of 7x - 2y - 3.
Since the slopes of parallel lines are the same, the slope of the other line is also 7/2.
General equation of the point-slope form:
The general equation of the point-slope form is given by:
y - y1 = m(x - x1), where
- (x1, y1) is any point on the line,
- and m is the slope.
Finding the equation of the line in point-slope form:
Now we can find the equation of the other line in point-slope form that passes through (-9, 5) and is parallel to the line whose equation is 7x - 2y - 3 = 0 by plugging in 7/2 for m and (-9, 5) for (x1, y1):
y - 5 = 7/2(x - (-9))
y - 5 = 7/2(x + 9)
Therefore, y - 5 = 7/2(x + 9) is the equation of the line in point-slope form passing through (-9, 5) and parallel to the line whose equation is 7x - 2y - 3 = 0.
Finding the equation of the line in general form:
Let's start by distributing 7/2:
y - 5 = (7/2 * x) + (7/2 * 9)
y - 5 = 7/2x + 63/2
Now we can clear the fractions by multiplying the equation by 2:
2(y - 5 = 7/2x + 63/2)
2y - 10 = 7x + 63
Now we can subtract 7x and 63 from both sides of the equation to find the equation of the line in general form:
(2y - 10 = 7x + 63) - 7x - 63
-7x + 2y - 73 = 0
Thus, -7x + 2y - 73 = 0 is the equation of the line in general form passing through (-9, 5) and parallel to the line whose equation is 7x - 2y - 3 = 0.