Questions on Algebra: Linear Equations, Graphs, Slope answered by real tutors!

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 Algebra: Linear Equations, Graphs, Slope Solvers Lessons Answers archive Quiz In Depth

Question 751147: Find all points on the y-axis that are 6 units from the point (4, -3). please help!
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1.Get a piece of graph paper.
2.Plot the point (4,-3).
3.Draw a horizontal line from (4,-3) connecting to the y-axis at the point (0,-3)
4.The line forms right angles with the y-axis and it's length is 4.
5.Now if you form a right triangle by drawing another line from point (4,-3) and connecting to the y-axis. The coordinates of where it connects to the y-axis will be (0, -3+a) and (0, -3-a). We are going to find a using the Pythagorean theorem, with b=4 and c=6.

The coordinates on the y-axis that are 6 units from (4,-3) are (0, -3+2sqrt(5)) and (0, -3-2sqrt(5))

Question 751154: System of two linear Equations in two variables
3x+ 2y= 4
4x- 3y= 11

You can put this solution on YOUR website!
Let's start by multiplying each equation so the y coefficient is 6.
3x+ 2y= 4 (multiply both side by 3)
4x- 3y= 11 (multiply both side by 2)
9x+ 6y= 12
8x- 6y= 22
17x+0y=34
x=2
3x+ 2y= 4
3(2)+2y=4
6+2y=4
2y=-2
y=-1
ANS x=2, y=-1

Question 751149: Company ABC’s profit is determined by the following equation: P = 30x + 40y, where x and y are sales of particular products and P is in thousands of dollars. It was determined that maximum and minimum profit values occur when sales occur in the following ordered pairs: (2,6), (6,4) and (-1,-3). At which pair is the maximum profit and at which pair is the minimum – state the profit.
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P = 30x + 40y
P(2,6)=30(2)+40(6)= 300
P(6,4)=30(6)+40(4)= 340
P(-1,-3)=30(-1)+40(-3)=-150

Max profit was at (6,4) and was $304,000 Min profit was at (-1,-3) and was a loss of$150,000

:)

Question 751047: A father opens a bank account when his daughter is born and deposits $5. Each year on her birthday he deposits twice the amount that he did in the previous year (i.e. on her first birthday he deposits$10). He continues this until she is 16 years old. Assuming that no interest is paid, how much is in the account after this time. This is a nonlinear equation but is also a nonlinear sequence can you help me find the Tn=?

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A father opens a bank account when his daughter is born and deposits $5. Each year on her birthday he deposits twice the amount that he did in the previous year (i.e. on her first birthday he deposits$10). He continues this until she is 16 years old.
5, 10, 20, 40, 80, 160, 320, 640, ....

(x(sub 1))=5 and any element in the sequence is 2(x(sub i -1)) where i = 2,3,...16
we want the summation of x(sub i) where i = 1,2,3,...16
observe that this is a Geometric progression, that is, the quotient of any two successive members of the sequence is a constant
in our sequence we see 10/5 = 2, 20/10 = 2, 40/20=2
for our problem the common ratio (q) is 2 and the scale factor is 5
formula for the sum of the first n numbers of a geometric series is
s(sub n) = x(sub 1) * (1 - q^n) / (1-q)
s(sub 16) = 5 * (1 - 2^16) / (1-2)
s(sub 16) = 5 * (1 - 65536) / (-1)
s(sub 16) = 5 * 65535
s(sub 16) = 327,675

Question 751076: Find the x- and y-intercepts for the line given by the equation 3x + 4y = 12

Question 750921: (-3,?) Is a solution of y=4-3x
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y=4-3x

y=4-3(-3) ... plug in x = -3

y=4 + 9

y=13

So (-3, 13) is a solution of y = 4 - 3x

Question 750869: Determine whether the given linear equations are parallel, perpendicular, or neither.
8x - 4y = - 12
8x - 4y = 8

Found 2 solutions by JoeTaxpayer, Cromlix:
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8x-4y=-12
-4y=-8x-12
y=2x+3 (first equation, rearranged. slope is 2, y intercept is 3)
8x - 4y = 8
2x -y = 2
-y=-2x+2
y=2x-2 (second equation rearranged. slope is 2, y intercept is -2)
Slopes are the same, but different y intercepts, so these lines are parallel.

You can put this solution on YOUR website!
Sort equations into y = mx + c
8x - 4y = - 12 => y = 2x + 3
8x - 4y = 8 => y = 2x - 2
As both lines have the same
are parallel to one another.
Hope this helps
:-)

Question 750820: Find the slope of the line 5x+2y=1
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Find the slope of the line 5x+2y=1
-----
2y = -5x + 1
-----
y = (-5/2)x + (1/2)
-----
slope = -5/2
===============
Cheers,
Stan H.
===============

Question 750816: Y intercept for 10x-20y=4500

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Y intercept for 10x-20y=4500
-----
x - 2y = 450
-----
2y = x-450
---
y = (1/2)x - 225
----
f(0) = -225 is the y-intercept
=================================
Cheers,
Stan H.
============

Question 750713: I need help with a sitprob that my teacher gave me. This'll be worth a good 20% of my term, being the 3rd term, it'll be worth 12%~ of my entire year. Please make sure you know what you're doing :)
I took a picture of it, wondering if anybody could help and give me steps.
http://www1.picturepush.com/photo/a/13114909/img/13114909.jpg
I already did the boys, not sure if it was done correctly or not though. I used the regression line method and seperated the boys into two equal groups (removed the last set of coords/results because they were uneven).
The girls part, if I didn't already mess up the boys, is where I'm having trouble. Please help me find the function/equation of the graph below.
Thankyou :)

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You have a complex and time-consuming problem here, time being valuable to us and getting this problem worked apparently being valuable to you. Write back and perhaps we can determine a mutually agreeable value.

John

Egw to Beta kai to Sigma
My calculator said it, I believe it, that settles it

Question 750644: What is the perimeter of the rectangle with the following corners: (5,2), (-5,2) and (5,-3)?
You can put this solution on YOUR website!
as per condition
the coordinate of vertex are given as (-5,2), (5,2) ,(5,-3) therefore it is given that it is rectangle
seeing conditions we can say fourths vertex coordinate will be (-5,-2)
by using distance formula we can say it will have length 10 cm and breath 5 cm
perimeter will be 2(l+b)
2(10 +5) =2x15 = 30 cm

Question 750518: Find the equation of the line (4,1)
You can put this solution on YOUR website!
please repost with more data like a slope or another point :)

Question 750504: Write the equation of the line that passes through (1,3) with a slope of 5.
Found 2 solutions by JoeTaxpayer, jim_thompson5910:
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slope point form is the way to solve Y-y1=m(X-x1)
Y-3=5(X-1)
Y-3=5X-5
Y=5X-2

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So the equation of the line is

Question 750502: Write the equation of the line that passes through (6, -2) with slope of -3.
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So the equation of the line is

Question 750430: graf y=x-2
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Looking at we can see that the equation is in slope-intercept form where the slope is and the y-intercept is

Since this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis

So we have one point

Now since the slope is comprised of the "rise" over the "run" this means

Also, because the slope is , this means:

which shows us that the rise is 1 and the run is 1. This means that to go from point to point, we can go up 1 and over 1

So starting at , go up 1 unit

and to the right 1 unit to get to the next point

Now draw a line through these points to graph

So this is the graph of through the points and

Question 750366: find the slope and the y-intercept of the straight line determined by the equation :3x+6y-12=0
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3x+6y-12=0
3x+6y=12
6y=-3x+12
y=-3/6x+2
y=-1/2x+2
y int =2
(0,2)
slope m=-1/2

Question 750264: how do you find the x intercept and slope of x= -5
____
3

You can put this solution on YOUR website!
is the equation of a vertical line. Every point
on the line is ( -5, y ), where can be any real number.
The slope is infinite, and the x-intercept is ( -5, 0 )

Question 750238: hello, need more help.
Find the Y-intercept and the X-intercept for the line determined by the points (1,-2) and (3,4). thank you for any help, I am looking for examples so I can start figuring this out. thanks again!

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 Solved by pluggable solver: Find the equation of line going through points hahaWe are trying to find equation of form y=ax+b, where a is slope, and b is intercept, which passes through points (x1, y1) = (1, -2) and (x2, y2) = (3, 4). Slope a is . Intercept is found from equation , or . From that, intercept b is , or . y=(3)x + (-5) Your graph:

since is slope-intercept form where is a slope and is a
so, is ; point (,)

: set to find

... is ; point (,)

Question 750228: Write an equation in slope-intercept form for the line that satisfies the following condition.
passes through (12, –4), perpendicular to the graph of y = 7/12x + 22

You can put this solution on YOUR website!
The given equation has slope =
The slope of any line perpendicular to this one
will have slope =

-----------------------
( 12, -4 )
Using the general point-slope formula:

-------------------------
check:
Does it go through ( 12, -4 ) ?

OK

Question 750229: (-5,-2),(3,-1) write and equation of the line that passes through the pair of points
You can put this solution on YOUR website!
( -5, -2 ) and ( 3, -1 )
The slope is ( change in y ) / ( change in x )

------------
Use the general point-slope formula

---------------------
check:
Does it go through (-5, -2) ?

OK
and
Does it go through ( 3,-1 )?

OK

Question 750230: I need help figuring out a problem please.
Point A is the origin, find the coordinates of the two points that are 5 units distant from A, and have an x-coordinate of +3. (The two lines should have coordinates of (3,y)). help please! thank you!

Found 3 solutions by KMST, rothauserc, Alan3354:
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The 2 points have y=4 and y=-4.
They are on the line .
Their coordinates are (3,-4) and (3,4).
Each one forms a right triangle with other vertices A(0,0) and (3,0)
The legs of those right triangles have lengths and , and the hypotenuse has length

You can put this solution on YOUR website!
we are given
Point A is the origin, find the coordinates of the two points that are 5 units distant from A, and have an x-coordinate of +3. (The two lines should have coordinates of (3,y))
this is a 3, 4, 5 right triangle
so the two points are
(3,4) and (3, -4)
the hypotenuse is 5 units which is the distance

You can put this solution on YOUR website!
Point A is the origin, find the coordinates of the two points that are 5 units distant from A, and have an x-coordinate of +3. (The two lines should have coordinates of (3,y)). They're 2 points, not 2 lines.
---------------
I did this one yesterday.

Question 750105: write an equation in slope-intercept form for the table of values:
X -3 -1 1 3
Y 7 5 3 1

Found 2 solutions by JoeTaxpayer, Alan3354:
You can put this solution on YOUR website!
2 points define a line. (1,3) (3,1) is what I'll use.
slope-intercept is the form y=mx+b where m is the slope and b is the y intercept.
m=(y2-y1)/(x2-x1)= (1-3)/(3-1) = -2/2 = -1
with the slope, we substitute a point into this equation -
y-3=-1(x-1)
y-3=-1x+1 (distribute the -1 over the two terms)
y=-1x+4 (add 3 to both sides to simplify)
Now - you should double check that the equation works for the extra two pairs we didn't use. To be sure there were no mistakes along the way, and no typo in the question)

You can put this solution on YOUR website!
write an equation in slope-intercept form for the table of values:
X -3 -1 1 3
Y 7 5 3 1
--------
Pick 2 of the points
(1,3) & (3,1) will do
Slope m = diffy/diffx
m = -2/2 = -1
---
Use y = mx + b and any point to find b, the y-intercept
(3,1)
1 = (-1)*3 + b
b = 4
-------
y = -x + 4

Question 750069: 8j-5=2j+13
You can put this solution on YOUR website!
8j-5=2j+13
Collect like terms
8j - 2j = 13 + 5
6j = 18
j = 3
:-)

Question 750010: Eight times a number is decreased by three times the same number, giving a difference of 20. What is the number?
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Hello,
Let's use x as the number:
8x-3x=20
Combine like terms:
5x=20
Divide by 5
x=4
Make sense?
RJ
www.math-unlock.com

Point A is the origin. Find the coordinates of the two points that are 5 units distant from A, and have x-coordinates of +3. (Hint: make a sketch. The two points fall on the line x=5, and therefore have coordinates (5,y) where y is unknown. Use the distance formula, suitably rearranged.)

You can put this solution on YOUR website!
Point A is the origin. Find the coordinates of the two points that are 5 units distant from A, and have x-coordinates of +3.
----------------
A is (0,0)
----

diffx = 3
----

y = -4, +4
--> (3,-4) and (3,4)

Question 749770: Find the x-intercepts and y-intercept of y = -x^2 + 3x + 4. Sketch the curve, using the x-intercepts and y-intercept to help you.
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Find y intercept using x=0. , so you have (0,4).

is the factored form of the given function which you can also rewrite as:
.
The value of y will be 0 when or when .
The x intercepts are then (-1,0) and (4,0). Note also, the maximum point occurs at , the parabola opens downward.

Compare that information to the graph:

Question 749683: which equation represents a line parellel to the line whose equation is 2y-5x=10?
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2y-5x=10
2y=5x+10
y=2.5X+5
This form, y=mx+b is a straight line, slope m and y intercept b.
All straight lines in the xy plane with a slope of 2.5 are parallel to this line. y=2.5x+b where b= any real number.

Question 749593: I have a question for "What is the equation of a line that has a slope of zero and a y-intercept of 4?"
and
"What is the equation of a line that has a slope of 3 and a y-intercept of -3?"

You can put this solution on YOUR website!
I have a question for "What is the equation of a line that has a slope of zero and a y-intercept of 4?"
y = 0x + 4
y = 4 is AN equation for the line.
-----------------
and
"What is the equation of a line that has a slope of 3 and a y-intercept of -3?
y = 3x - 3 is AN equation for the line.

Two stations A and B are 90km apart and on a highway.A car starts from A and another from B at the same time.If they go in the same direction, they meet after 9 hours and if they go in opposite directions,they meet after 1 2/7 (one- two by seven hours)hours, find the speeds of both the cars.

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Two stations A and B are 90km apart and on a highway.
A car starts from A and another from B at the same time.
If they go in the same direction, they meet after 9 hours and
if they go in opposite directions, they meet after 1 2/7 hours,
find the speeds of both the cars.
:
Change 1 2/7 hrs to 9/7 hrs
:
Let x = Car from A's speed
Let y = Car from B's speed
:
A distance equation when going towards each other
x + y = 90
multiply by 7
9x + 9y = 630
:
A distance equation when going the same direction
9x = 9y + 90
9x - 9y = 90
:
Use elimination here
9x + 9y = 630
9x - 9y = 90
18x = 720
x = 720/18
x = 40 km/hr speed of the Car from A
then
9(40) + 9y = 630
9y = 630 - 360
9y = 270
y = 270/9
y = 30 km/hr speed of the car from B
:
:
You can confirm this
9(40) - 9(30) = 90

Question 749453: Which is an equation of the line that passes through the point (2, 0) and is parallel to the graph of y = –4x + 1?
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The full equation you give is of the form y=mx+b where m is the slope.
Here, it's -4
Simply put, parallel lines have the same slope.
So we need to create a line with slope -4 that passes through (2,0) so we use the form:
y-0=-4(x-2)
ANS y=-4x+8

Question 749525: On Tuesday the express train made the trip in 12 hours. On Wednesday the freight train made the trip in 16 hours. Find the rate of each train if the freight train was 15 miles per hour slower than the express train.
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On Tuesday the express train made the trip in 12 hours. On Wednesday the freight train made the trip in 16 hours. Find the rate of each train if the freight train was 15 miles per hour slower than the express train.
***
let x=distance traveled
rate=distance/travel time
x/12=rate of express train
x/16=rate of freight train
..
x/12-x/16=15
LCD=12*16=192
16x-12x=2880
4x=2880
x=720
x/12=60
x/16=45
rate of express train=60 mph
rate of freight train=45 mph
..

1.y=-3x-6;(-1,5) Write the equation of a line parallel to the given line but passing through the given point.
2. y=-1/3x-4;(-6,-3)Write the equation of a line perpendicular to the given line but passing through the given point.

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1.y=-3x-6;(-1,5) Write the equation of a line parallel to the given line but passing through the given point.
Slope of the given equation is -3, so the parallel line has slope = -3.
--------
Form: y = mx + b
Solve for "b":
5 = -3*-1 + b
b = 2
----
Equation:
y = -3x + 2
================
2. y=-1/3x-4;(-6,-3) Write the equation of a line perpendicular to the given line but passing through the given point.
-------
The given line has slope -1/3, so the perpendicular line has slope 3.
Form: y = mx + b
-3 = 3*-6 + b
-3 = -18 + b
b = 15
-----
Equation:
y = 3x + 15
=================
Cheers,
Stan H.
=================

Question 749456: What is 2+2
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What is 2+2

Question 749384: A line with a slope of -(1/2) crosses the X-axis at the point (4,0). At what point does it cross the Y-axis? (Hint: The unknown point has coordinates (0,y), where y is to be calculated. This point is the so-called y intercept.)
You can put this solution on YOUR website!
Using y - b = m(x - a)
y - 0 = -1/2(x - 4)
y = -1/2x + 2
Where the line cuts the y axis
x = 0
Therefore y = -1/2(0) + 2
y = 2
So the coordinates (0, 2) is
where it cuts the y axis

Question 749291: Find the x-intercept and y-intercept of the line given by the equation.
6x+8y=12

You can put this solution on YOUR website!
Find the x-intercept and y-intercept of the line given by the equation.
6x+8y=12
x-intercept ?
Let y = 0, then x = 2
-----
y-intercept ?
Let x = 0, then y = 12/8 = 3/2
==========================================
Cheers,
Stan H.

Question 749252: I need to find the y intercept and gradient for this. Please give steps to calculate. 2x-5y=10
Found 2 solutions by Cromlix, solver91311:
You can put this solution on YOUR website!
2x - 5y = 10
Put items in the form
y = mx + c
m= gradient c = y intercept
-5y = -2x + 10
Multiply through by -1
5y = 2x - 10
Divide through by 5
y = 2/5x - 2
Gradient = 2/5 and y intercept
is -2

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Multiply both sides by

The result will be in the form

The value of is the slope or gradient, and the point is the -intercept.

John

Egw to Beta kai to Sigma
My calculator said it, I believe it, that settles it

Question 749230: A 15 foot ladder is leaning against a house with its bottom 9 feet from the base of the house. What is the slope of the ladder?
You can put this solution on YOUR website!
Trigonometric ratio.
First find the remaining third
side.
15^2 - 9^2 = 144
square root of 144 = 12
= 12/9
x = 53.1 degrees
Slope = 1.3

Question 749199: y-4=-1/3(x-2)
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Do you have a question?

Question 749054: write an equation in slope intercept form for the line that satisfies the following condition x-intercept 12 and y-intercept 14
You can put this solution on YOUR website!

To begin with, your terminology is incorrect. Intercepts are points and points (in ) are specified using ordered pairs. So giving a number and calling it an intercept is incorrect. Presuming you mean that the -coordinate of the -intercept is 12 and the -coordinate of the -intercept is 14, then you have been given two points on the desired line, namely and .

Use the two-point form:

where and are the coordinates of the given points.

Once you have made the substitutions of the coordinates, simplify and solve for in terms of everything else. That is to say, put your equation in the form:

John

Egw to Beta kai to Sigma
My calculator said it, I believe it, that settles it

Older solutions: 1..45, 46..90, 91..135, 136..180, 181..225, 226..270, 271..315, 316..360, 361..405, 406..450, 451..495, 496..540, 541..585, 586..630, 631..675, 676..720, 721..765, 766..810, 811..855, 856..900, 901..945, 946..990, 991..1035, 1036..1080, 1081..1125, 1126..1170, 1171..1215, 1216..1260, 1261..1305, 1306..1350, 1351..1395, 1396..1440, 1441..1485, 1486..1530, 1531..1575, 1576..1620, 1621..1665, 1666..1710, 1711..1755, 1756..1800, 1801..1845, 1846..1890, 1891..1935, 1936..1980, 1981..2025, 2026..2070, 2071..2115, 2116..2160, 2161..2205, 2206..2250, 2251..2295, 2296..2340, 2341..2385, 2386..2430, 2431..2475, 2476..2520, 2521..2565, 2566..2610, 2611..2655, 2656..2700, 2701..2745, 2746..2790, 2791..2835, 2836..2880, 2881..2925, 2926..2970, 2971..3015, 3016..3060, 3061..3105, 3106..3150, 3151..3195, 3196..3240, 3241..3285, 3286..3330, 3331..3375, 3376..3420, 3421..3465, 3466..3510, 3511..3555, 3556..3600, 3601..3645, 3646..3690, 3691..3735, 3736..3780, 3781..3825, 3826..3870, 3871..3915, 3916..3960, 3961..4005, 4006..4050, 4051..4095, 4096..4140, 4141..4185, 4186..4230, 4231..4275, 4276..4320, 4321..4365, 4366..4410, 4411..4455, 4456..4500, 4501..4545, 4546..4590, 4591..4635, 4636..4680, 4681..4725, 4726..4770, 4771..4815, 4816..4860, 4861..4905, 4906..4950, 4951..4995, 4996..5040, 5041..5085, 5086..5130, 5131..5175, 5176..5220, 5221..5265, 5266..5310, 5311..5355, 5356..5400, 5401..5445, 5446..5490, 5491..5535, 5536..5580, 5581..5625, 5626..5670, 5671..5715, 5716..5760, 5761..5805, 5806..5850, 5851..5895, 5896..5940, 5941..5985, 5986..6030, 6031..6075, 6076..6120, 6121..6165, 6166..6210, 6211..6255, 6256..6300, 6301..6345, 6346..6390, 6391..6435, 6436..6480, 6481..6525, 6526..6570, 6571..6615, 6616..6660, 6661..6705, 6706..6750, 6751..6795, 6796..6840, 6841..6885, 6886..6930, 6931..6975, 6976..7020, 7021..7065, 7066..7110, 7111..7155, 7156..7200, 7201..7245, 7246..7290, 7291..7335, 7336..7380, 7381..7425, 7426..7470, 7471..7515, 7516..7560, 7561..7605, 7606..7650, 7651..7695, 7696..7740, 7741..7785, 7786..7830, 7831..7875, 7876..7920, 7921..7965, 7966..8010, 8011..8055, 8056..8100, 8101..8145, 8146..8190, 8191..8235, 8236..8280, 8281..8325, 8326..8370, 8371..8415, 8416..8460, 8461..8505, 8506..8550, 8551..8595, 8596..8640, 8641..8685, 8686..8730, 8731..8775, 8776..8820, 8821..8865, 8866..8910, 8911..8955, 8956..9000, 9001..9045, 9046..9090, 9091..9135, 9136..9180, 9181..9225, 9226..9270, 9271..9315, 9316..9360, 9361..9405, 9406..9450, 9451..9495, 9496..9540, 9541..9585, 9586..9630, 9631..9675, 9676..9720, 9721..9765, 9766..9810, 9811..9855, 9856..9900, 9901..9945, 9946..9990, 9991..10035, 10036..10080, 10081..10125, 10126..10170, 10171..10215, 10216..10260, 10261..10305, 10306..10350, 10351..10395, 10396..10440, 10441..10485, 10486..10530, 10531..10575, 10576..10620, 10621..10665, 10666..10710, 10711..10755, 10756..10800, 10801..10845, 10846..10890, 10891..10935, 10936..10980, 10981..11025, 11026..11070, 11071..11115, 11116..11160, 11161..11205, 11206..11250, 11251..11295, 11296..11340, 11341..11385, 11386..11430, 11431..11475, 11476..11520, 11521..11565, 11566..11610, 11611..11655, 11656..11700, 11701..11745, 11746..11790, 11791..11835, 11836..11880, 11881..11925, 11926..11970, 11971..12015, 12016..12060, 12061..12105, 12106..12150, 12151..12195, 12196..12240, 12241..12285, 12286..12330, 12331..12375, 12376..12420, 12421..12465, 12466..12510, 12511..12555, 12556..12600, 12601..12645, 12646..12690, 12691..12735, 12736..12780, 12781..12825, 12826..12870, 12871..12915, 12916..12960, 12961..13005, 13006..13050, 13051..13095, 13096..13140, 13141..13185, 13186..13230, 13231..13275, 13276..13320, 13321..13365, 13366..13410, 13411..13455, 13456..13500, 13501..13545, 13546..13590, 13591..13635, 13636..13680, 13681..13725, 13726..13770, 13771..13815, 13816..13860, 13861..13905, 13906..13950, 13951..13995, 13996..14040, 14041..14085, 14086..14130, 14131..14175, 14176..14220, 14221..14265, 14266..14310, 14311..14355, 14356..14400, 14401..14445, 14446..14490, 14491..14535, 14536..14580, 14581..14625, 14626..14670, 14671..14715, 14716..14760, 14761..14805, 14806..14850, 14851..14895, 14896..14940, 14941..14985, 14986..15030, 15031..15075, 15076..15120, 15121..15165, 15166..15210, 15211..15255, 15256..15300, 15301..15345, 15346..15390, 15391..15435, 15436..15480, 15481..15525, 15526..15570, 15571..15615, 15616..15660, 15661..15705, 15706..15750, 15751..15795, 15796..15840, 15841..15885, 15886..15930, 15931..15975, 15976..16020, 16021..16065, 16066..16110, 16111..16155, 16156..16200, 16201..16245, 16246..16290, 16291..16335, 16336..16380, 16381..16425, 16426..16470, 16471..16515, 16516..16560, 16561..16605, 16606..16650, 16651..16695, 16696..16740, 16741..16785, 16786..16830, 16831..16875, 16876..16920, 16921..16965, 16966..17010, 17011..17055, 17056..17100, 17101..17145, 17146..17190, 17191..17235, 17236..17280, 17281..17325, 17326..17370, 17371..17415, 17416..17460, 17461..17505, 17506..17550, 17551..17595, 17596..17640, 17641..17685, 17686..17730, 17731..17775, 17776..17820, 17821..17865, 17866..17910, 17911..17955, 17956..18000, 18001..18045, 18046..18090, 18091..18135, 18136..18180, 18181..18225, 18226..18270, 18271..18315, 18316..18360, 18361..18405, 18406..18450, 18451..18495, 18496..18540, 18541..18585, 18586..18630, 18631..18675, 18676..18720, 18721..18765, 18766..18810, 18811..18855, 18856..18900, 18901..18945, 18946..18990, 18991..19035, 19036..19080, 19081..19125, 19126..19170, 19171..19215, 19216..19260, 19261..19305, 19306..19350, 19351..19395, 19396..19440, 19441..19485, 19486..19530, 19531..19575, 19576..19620, 19621..19665, 19666..19710, 19711..19755, 19756..19800, 19801..19845, 19846..19890, 19891..19935, 19936..19980, 19981..20025, 20026..20070, 20071..20115, 20116..20160, 20161..20205, 20206..20250, 20251..20295, 20296..20340, 20341..20385, 20386..20430, 20431..20475, 20476..20520, 20521..20565, 20566..20610, 20611..20655, 20656..20700, 20701..20745, 20746..20790, 20791..20835, 20836..20880, 20881..20925, 20926..20970, 20971..21015, 21016..21060, 21061..21105, 21106..21150, 21151..21195, 21196..21240, 21241..21285, 21286..21330, 21331..21375, 21376..21420, 21421..21465, 21466..21510, 21511..21555, 21556..21600, 21601..21645, 21646..21690, 21691..21735, 21736..21780, 21781..21825, 21826..21870, 21871..21915, 21916..21960, 21961..22005, 22006..22050, 22051..22095, 22096..22140, 22141..22185, 22186..22230, 22231..22275, 22276..22320, 22321..22365, 22366..22410, 22411..22455, 22456..22500, 22501..22545, 22546..22590, 22591..22635, 22636..22680, 22681..22725, 22726..22770, 22771..22815, 22816..22860, 22861..22905, 22906..22950, 22951..22995, 22996..23040, 23041..23085, 23086..23130, 23131..23175, 23176..23220, 23221..23265