4x-9
4x+6
Can anybody explain how to get the answer clearly and preferably with a step by step process?For 10 POINTS - For A Clear Math Explanation....?
How far are -9 and 6 apart? 15.
That's the answer. The line y = x + 6 is just the line y = x - 9 shifted 15 units up along the y-axis.
-JohnFor 10 POINTS - For A Clear Math Explanation....?
The shortest distance between two parallel lines is perpendicular to those lines. You know from y=mx+b that the slope (m) is 4. The way you find a perpendicular slope is to take the negative inverse of that slope. So in this case -1/4. Now imagine we draw a line perpendicular at the y-intercept (0,-9). Using the point-slope equation for a line we find the equation of the perpendicular line to be:
(y-y1)=m(x-x1)
(y--9)=-1/4(x-0)
y+9=-1/4x
y=-1/4x-9
Now we find the intersection between the top parallel line and the perpendicular line:
y=4x+6
y=-1/4x-9
0=17/4x+15
x=-60/17
y=4(-60/17)+6
y=-240/17+6
y=-138/17
(-60/17,-138/17)
Now all we have to do is find the distance between those points using pythagorean theorem:
d=sqrt(a^2+b^2)
d=sqrt((-60/17-0)^2+(-138/17--9)^2)
d=sqrt(3600/289+225/289)
d=sqrt(3825/289)
d=sqrt(3825)/17
d=15*sqrt(17)/17
d=3.638
you need to find two points that any arbitrary perpendicular line intersects the two parallel lines and then find the distance between those two points:
a perpendicular line is of the form:
y=-x/m + b
in this case, m=4
choose b=-9 for convience to eliminate the -9 from the first equation:
y=-x/4 - 9 is the perpendicular line
set the two equal to find where they intersect:
-x/4 - 9 = 4x - 9
so x=0
sub that in to get y
4(0)-9=-9
so one point is (0,-9)
now set the perpendicular line to the other equation:
-x/4 - 9 = 4x + 6
-15 = 15x/4
x=-4
sub back in to get y:
4(-4)+6=-10
so the other point is:
(-4,-10)
the distance between the two points is:
d=sqrt((0-(-4))^2 + (-9-(-10))^2)
=sqrt(4^2 + 1^2)
=sqrt(17)
it is not 15, that is just the vertical distance. The actual distance is the minimum of one of the lines and a fixed point on the other line. This means that another way to solve this would be to find a point on one of the lines and then to find the distance between that point and the other line and then minimize that:
if x=0
4(0)+6=6
so the point is (0,6)
the distance between that point and the other line is:
d=sqrt((x-(0))^2 + (y-6)^2)
=sqrt(x^2 + ((4x-9)-6)^2)
=sqrt(x^2 + (4x-15)^2)
=sqrt(17x^2 -120x + 225)
now take the derivative and set to 0 to find minimum:
(1/2)(17x^2 - 120x +225)^(-1/2) (34x - 120)
The answer is much less than 15. The answerer who said the distance will be along the perpendicular line between them is correct. Because the slope is 4, these lines will be fairly close together (because they're fairly steep, the difference in y-intercepts will be less important). To convince yourself of this, draw a quick sketch of these two lines. They are much closer together than 15 units! They would be 15 units apart ONLY if they were horizontal lines with y-intercepts of 6 and -9.
y = 4x - 9
y = 4x + 6
both have a slope of 4, so the perpendicular line will have a slope of -1/4
now pick a point on one line to find the equation of a perpendicular line through that point.
We'll use (0,6) on the second line, which is the y-intercept.
Therefore the perpendicular line through this point has slope
-1/4 and y-intercept 6
equation is y=(-1/4)x + 6
find the intersection of this line with the first line by setting the two equations equal.
y = 4x - 9 = (-1/4)x + 6
(4+1/4)x = 6 + 9
17/4 x = 15
x = 60/17
when x = 60/17, y = 4(60/17) - 9 = 240/17 - 153/17 = 87/17
so the point of intersection is (60/17,87/17)
The distance between the lines is the distance between (0,6) and (60/17 , 87/17) (which are the two points of intersection with a perpendicular line)
Using the distance formula, and converting to decimals:
(0,6) and (3.53, 5.12)
d = sqrt[3.53^2 + (6 - 5.12)^2] = sqrt(13.23) = 3.64
The shortest distance between these lines is 3.64 units
Note: Josh E has the right idea below, also, but made an arithmetic in this step:
-x/4 - 9 = 4x + 6
-15 = 15x/4 %26lt;====== should be -15 = 17x/4
x = -60/17
with this correction, we have the same answer!
I'm sure you've seen this equation:
y=mx+b
The 'b' is the y-intercept. It tells you where this line crosses the y-axis. On the first line, it crosses at -9. On the second, it crosses at 6. It might help to get some graph paper and draw both lines. If so, you could just count the spaces between the lines. It's easier to visualize this way.
In any case, by subtracting these numbers, we know that the lines are spaced 15 units apart. The first one is sitting 9 spaces below (0,0) and the second is sitting 6 units over (0,0).
*the slop formula is y=mx+b*
M is the change of the slop, B is the y-intercept.
*If both m's (4x) are the same the slop does not change.
*the y-intercept changes by 15 spaces, think about the y-axis on the graph, 9 spaces down, then 6 spaces up. OR use a line, then label, count how many spaces there is b/w the 2 lines.
The answer you want is : 15 spaces.
ok you now that 4x = 4x so they mark each other out. so then you just subtract -9 - 6 = -15 . but you are wanting to know the distance between each line so you would find the absolute value (the distance away from 0) of -15 which is 15 so your answer is 15.
These are parallel lines. Look at the y intercepts. The first line crosses the y axis at -9 while the second crosses at 6. Shortest distance between two lines will be perpindicular distance. Hence the distance is |9| + |6| =15 units
The first line is shifted down 9 units on the y-axis.
The second line is shifted up 6 units on the x-axis.
So they are 15 units apart.
You could also use subtraction to calculate the ';difference';
6 - (-9)
6 + 9
15
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