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This is one of the problems submitted. Reference to which one is deliberately removed to prevent easy copying.

Write a slope-intercept equation for a line passing through the given point that is parallel to the given line. Then write a second equation for a line passing through the given point that is perpendicular to the given line.

(-4,-5), 2x + y = -4

This problem splits into five steps which are broken down below.

1. Put the equation to slope-intercept form and determine the slope.
2. Write a parallel equation.
3. Write a perpendicular equation.
4. Summary.
5. Check.

1.A. Put the original equation in slope-intercept form of y = mx + b.

2x + y = -4
     y = -4 - 2x
     y = -2x - 4
     y = -2x + (-4)

1.B. Determine slope from slope-intercept equation.

y = -2x + (-4)
m = -2

2.A. Write a parallel equation using the slope.

y = -2x + b

2.B. Find the y-intercept for the parallel line that goes through the given point.

Given point: (-4,-5)

So:
x = -4
y = -5

Equation:
 y     = -2x   + b

Substite x and y:
-5     = -2*-4 + b
-5     = 8     + b
-5 - 8 =         b
-13    =         b
b = -13

2.C. Using the slope and y-intercept, write parallel equation.

y = -2x + (-13)
y = -2x - 13

3.A. Using the slope, find a perpendicular slope using the formula m₁m₂ = -1.

Parallel slope (from above):
m = -2
m1 = -2

 m1 * m2 = -1
-2  * m2 = -1
      m2 = -1/-2
      m2 = 1/2

So, perpendicular slope:
m = 1/2

3.B. Using the perpendicular slope, write a perpendicular equation.

y = (1/2)x + b

3.C. Find the y-intercept for the perpendicular line that goes through the given point.

Given point: (-4,-5)

So:
x = -4
y = -5

Equation:
 y     = (1/2)x   + b

Substite x and y:
-5     = (1/2)*-4 + b
-5     = -4/2     + b
-5     = -2       + b
-5 + 2 =            b
b = -3

3.D. Using the slope and y-intercept, write perpendicular equation.

y = (1/2)x + b
y = (1/2)x + (-3)
y = (1/2)x - 3

Original given point and equation:
1.4. 60. (-4,-5), 2x + y = -4

Original equation in slope-intercept form:
y = -2x - 4

Parallel equation through given point:
y = -2x - 13

Perpendicular equation through given point:
y = (1/2)x - 3

5. Check. - Graph attached, graph data below, all color coded as above.

 

Formula Calculated y-intercept, by substituting 0 for x Calculated x-intercept, by substituting 0 for y Random x,y from graph, chosen from an intersection Substitute and solve random point

Original: y = -2x - 4 (0,-4) (-2,0) (-1,-2) -2 = -2(-1) - 4 -2 = 2 - 4 -2 = -2

Parallel: y = -2x - 13 (0,-13) (-6.5,0) (-3,-7) -7 = -2(-3) - 13 -7 = 6 - 13 -7 = -7

Perpendicular: y = (1/2)x - 3 (0,-3) (6,0) (2,-2) -2 = (1/2)*2 - 3 -2 = (1/2)*(2/1) - 3 -2 = 1 - 3 -2 = -2

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