Maths coordinate geometry help

the line L has equation 2x - y - 1 = 0. The line M passes through the point A(0,4) and is perpendicular to the line L.

1). find an equation of M and show that the lines L and M intersect at the point P(2,3)

& I can't do this either...

the line n passes through the point 3,0 and is parallel to the line M

2). find an equation of N and hence find the coordinates of the point Q where the lines L and N intersect
& 3). prove that AP=BQ=PQ

I really struggle with these kind of questions so if someone has the time to explain each question line by line i would be extremely grateful! thanks so much to anyone who helps :)

the given equation must be rearranged into the form y= mx+c so that the slope m can be seen

gives

-y = 1 - 2x
so
y= 2x - 1

the slope of the given line is the number in front of the x
slope = +2

Any line perpendicular to this will have a slope found as follows

If the first slope is m1 and the perpendicular slope is m2
then m1 times m2 = -1.................formula for perpendicular slopes

so m2 = -1 /m1
m2 = -1/+2 = -1/2

The general equation for any line is y = mx + c

we can now put in the m value for the perpendicular line

y = -x/2 + c

now we have to find the appropriate c value
we are given that when x = 0 y = 4

4 = -0/2 + c
so c = +4
The equation of line L is

y = 4 - x/2 or 2y = 8 - x

equation M

2x-y -1 = 0
equation L
2y = 8-x

They intersect at a point where their x and y value is the same

so solve as simultaneous equations

from equation L
y = 4 - x/2....................put this value for y in M
so
2x -[4 - x/2] - 1 = 0
2x - 4 + x/2 -1 = 0
5x/2 = 5
5x = 10
x = 2...........put this value for x in either equation to find y
y = 3

The point where they intersect [2,3]

a line that is parallel has the same slope as the line it is parallel to
[draw 2 parallel lines and you can see they slope the same]

so we know the slope is + 2
again using the general equation

y = 2x + c................now we need to find the c for our parallel line
when x = 3 y = 0........we are given this

so

0 = 6 + c................so our new c is - 6
go back to the general equation and we now have

y = 2x - 6............this is the equation of the parallel line

So the next bit you have to solve the simultaneous equations of L and N to find the x and y value as done above

have a go !!

noo