write the slope intercept form of the equation of the line with the following conditions.
19. passing through (4,-2) and (-8, -5).
21. passing through (-5,9) and parallel to 3x+y=1.
23. passing through (4,1) and perpendicular to 2x-y=8.
19. the slope is (-5+2)/(-8-4) = 1/4
so, the line is
y+2 = 1/4 (x-4)
or
y+5 = 1/4 (x+8)
Now use that same form for the others, remembering that the slopes of parallel lines are equal, and that the slopes of perpendicular lines multiply to -1.
To find the slope-intercept form of a line, we need two pieces of information: the slope and the y-intercept. Here's how you can find the slope-intercept form for each of the given conditions:
19. To find the slope, use the formula:
slope (m) = (y2 - y1) / (x2 - x1).
Let's use the points (4, -2) and (-8, -5) to calculate the slope:
m = (-5 - (-2)) / (-8 - 4)
= (-5 + 2) / (-12)
= -3 / -12
= 1/4
Now, we can use the point-slope form of a line (y - y1) = m(x - x1) and substitute the values of the slope and one of the given points:
(y - (-2)) = (1/4)(x - 4)
Simplifying this equation gives us:
y + 2 = (1/4)(x - 4)
y + 2 = (1/4)x - 1
y = (1/4)x - 3
The slope-intercept form of the line passing through (4, -2) and (-8, -5) is y = (1/4)x - 3.
21. The equation 3x + y = 1 is in standard form (Ax + By = C). To find a line parallel to this one, it must have the same slope. Therefore, we need to convert the given equation into slope-intercept form:
y = -3x + 1.
So, the slope of the line is -3.
Now, we can use the point-slope form to find the equation:
y - y1 = m(x - x1)
Using the point (-5, 9), we have:
y - 9 = -3(x - (-5))
y - 9 = -3(x + 5)
y - 9 = -3x - 15
y = -3x - 6
The slope-intercept form of the line passing through (-5, 9) and parallel to 3x + y = 1 is y = -3x - 6.
23. Similarly, let's convert the given equation 2x - y = 8 into slope-intercept form.
Subtracting 2x from both sides, we get:
-y = -2x + 8
Multiplying both sides by -1, we get:
y = 2x - 8.
So, the slope of this line is 2.
Now, we use the point-slope form with the given point (4, 1):
y - y1 = m(x - x1)
y - 1 = 2(x - 4)
y - 1 = 2x - 8
y = 2x - 7
The slope-intercept form of the line passing through (4, 1) and perpendicular to 2x - y = 8 is y = 2x - 7.