Find an equation of the line passing through the point (−1, 9) and parallel to the line joining the points (−3, 4) and

(4, −1).

the line through (−3, 4) and (4, −1) has slope -5/7

Any line oarallel to it has the same slope.
So, now you have a point and a slope, so the line you want is
y-9 = -5/7 (x+1)

To find the equation of the line passing through the point (-1, 9) and parallel to the line joining the points (-3, 4) and (4, -1), we need to determine the slope of the given line and use it to write the equation in the point-slope form.

The slope of a line passing through two points (x1, y1) and (x2, y2) can be calculated using the formula:

m = (y2 - y1) / (x2 - x1)

Let's calculate the slope of the given line:
m = (-1 - 4) / (4 - (-3))
= (-5) / (4 + 3)
= (-5) / 7

Since the line we want to find is parallel to this line, it will have the same slope. So the slope of the new line is -5/7.

Now we have the slope (-5/7) and a point (-1, 9). We can use the point-slope form of an equation to write the equation of this line:

y - y1 = m(x - x1)

where (x1, y1) is the given point and m is the slope.

Substituting the values, we get:

y - 9 = (-5/7)(x - (-1))

Simplifying,

y - 9 = (-5/7)(x + 1)

y - 9 = (-5/7)x - 5/7

Adding 9 to both sides:

y = (-5/7)x - 5/7 + 63/7

y = (-5/7)x + 58/7

Therefore, the equation of the line passing through the point (-1, 9) and parallel to the line joining the points (-3, 4) and (4, -1) is y = (-5/7)x + 58/7.

To find the equation of a line parallel to another line, we need to determine the slope of the given line.

The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:

m = (y2 - y1) / (x2 - x1)

In this case, the given line passes through the points (-3, 4) and (4, -1). So, we can calculate the slope of the line as follows:

m1 = (-1 - 4) / (4 - (-3))
= -5 / 7

Since our new line is parallel to the given line, it will have the same slope. Therefore, the slope of the new line is also -5/7.

Now, we have the slope of the line (-5/7) and a point it passes through (-1, 9). We can use the point-slope form of the equation of a line to find the equation of the new line:

y - y1 = m(x - x1)

where (x1, y1) is the given point and m is the slope.

Substituting the values, we get:

y - 9 = (-5/7)(x - (-1))
y - 9 = (-5/7)(x + 1)
Multiplying both sides by 7 to eliminate the fraction:
7(y - 9) = -5(x + 1)
7y - 63 = -5x - 5
7y = -5x + 58

Therefore, the equation of the line passing through the point (-1, 9) and parallel to the line joining the points (-3, 4) and (4, -1) is 7y = -5x + 58.