*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.

- Put the equation to slope-intercept form and determine the slope.
- Write a parallel equation.
- Write a perpendicular equation.
- Summary.
- 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 m_{1}m_{2} = -1.

Parallel slope (from above): m = -2 m_{1}= -2 m_{1}* m_{2}= -1 -2_{ }* m_{2}= -1_{ }m_{2}= -1/-2_{ }m_{2}= 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

4. Summary

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 |