Consider the line y = 3x-2.
(a) find the equation of the line that is parallel to this line and passes through the point (4, 4).
(b) find the equation of the line that is perpendicular to this line and passes through the point (4, 4).

Step-by-step explanation: (a)  y = 3x - 8 (b)  3y + x - 16 = 0

(a) The line is y = 3x - 2

But the condition for parallelism is that for two lines to be parallel to each other, their gradients m must be equal, ie, m1 = m2

therefore, the gradient of the line above m1 = 3, m2 = 3

since the line passes through the coordinate of ( 4, 4 ),

we need to find the y intersect ( c ) by substitute for x, m and y in the equation below.

y = mx  + c

4 = 3 x 4 + c

4 = 12  + c

c = 4 - 12

c = -8

Therefore, substitute for  c in the equation of a line above to get the second equation

y = mx + c

y = 3x - 8

(b) Condition for perpendicularity of two line is that  the product of their gradients must be( -1 )

ie, m1m2 = -1

Now from the equation above, y = 3x - 2, m1 = 3 and m2 = -1/3

to get the value of c, we substitute for  x, y and m into the equation

y = mx + c

4 = -1/3 x 4 = c

4 = -4/3 + c

multiply through by 3 to make it a linear equation

12 = -4 + 3c

12 + 4 = 3c

16 = 3c

c = 16/3

now put c = 16/3 into the equation , y=mx + c

y = -x/3  + 16/3

multiply through by 3

3y = -x + 16

3y + x - 16 =0

- "runner b is moving at a slower rate."

- "the graphs shown are graphs of distance(km) against time(min).

the slopes of this graphs are are given by:

slope = (change in y)/(change in x)

= (change in distance)/(change in time)

this is the formula for finding the speed. so, the slope of these graphs represent the speed of the runner.

taking each division in the graph to represent 1 on both axis:

runner a ⇒ speed = (2-0)/(1-0)

= 2/1

= 2 km/min

runner b ⇒ speed = (3-0)/(3-0)

= 3/3

= 1 km/min

this shows that   runner b is moving at a slower rate."