Answer:
(a) Slope: 2
(b) y = 2x - 5
(c) y = (-1/2)x + 8
Step-by-step explanation:
(a) To find the slope of the given line, we can express the linear equation in slope-intercept form: y = mx + b.
y: y coordinate
x: x coordinate
m: slope
b: y-intercept
In order to express our equation in slope-intercept form, we must solve the linear equation for y.
-8x + 4y = -24
Add 8x to both sides: 4y = 8x - 24
Divide the equation by 4: y = 2x - 6
We find that the slope of the line is 2.
(b) Parallel lines have the same slope. Since we found the slope of the first line to be 2, our parallel line must also have a slope of 2.
y = 2x + b
Now, using the point (4, 3), we can solve for b.
3 = 2(4) + b ⇒ 3 = 8 + b
Subtract both sides by 8: b = -5
Therefore the equation of the line parallel to the given line that goes through (4, 3) is y = 2x - 5.
(c) The slope of a line perpendicular to the given line is the negative reciprocal of the slope of the given line.
Since the slope of the given line is 2, the slope of the line perpendicular to it must be -1/2.
y = (-1/2)x + b
Now, using the point (8,4), we can solve for b.
4 = (-1/2)8 + b ⇒ 4 = -4 + b
Add 4 to both sides: b = 8
Therefore the equation of the line perpendicular to the given line that goes through (8, 4) is y = (-1/2)x + 8.
(a) Slope: 2
(b) Parallel: y = 2x - 5
(c) Perpendicular: y = [tex]-\frac{1}{2}[/tex]x + 8
(a) To find the slope, we will rewrite the given equation in slope-intercept form. In slope-intercept form, the slope of the equation is m in y = mx + b.
  Given:
     –8x + 4y = –24
  Add 8x to both sides of the equation:
     4y = 8x - 24
  Divide both sides of the equation by 4:
     y = 2x - 12
     Slope = 2.
(b) A parallel line will have the same slope as the given line. This is 2. Next, we are given the coordinate point (4, 3). With this information, we will write a point-slope form equation and transform it into slope-intercept form.
  Point-slope form equation:
     y - y1 = m(x - x1)
  Substitute known values:
     y - 3 = 2(x - 4)
  Distribute the 2 and add 3 to both sides of the equation:
     y = 2x - 8 + 3
  Add 3 to -8:
     y = 2x - 5
     The parallel line is y = 2x - 5.
(c) Lastly, the perpendicular line to the given line will have a negative reciprocal as its slope. Since our given line has a slope of 2, the perpendicular line will have a slope of -1/2. Next, we are given the coordinate point (8, 4). Again, we will use this information to create a point-slope form equation.
  Point-slope form equation:
     y - y1 = m(x - x1)
  Substitute known values:
     y - 4 =  [tex]-\frac{1}{2}[/tex](x - 8)
  Distribute and add 4 to both sides of the equation:
     y =  [tex]-\frac{1}{2}[/tex]x + 4 + 4
  Add 4 to 4:
     y =  [tex]-\frac{1}{2}[/tex]x + 9
     The perpendicular line is [tex]y = -\dfrac{1}{2} x + 8[/tex].
