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The price of an item is given by p=2x^2 - 100
Find the polynomial that represents the revenue generated from the sale of x items.
;
They don't want to solve it, just write an equation that will give you the amt of revenue for a given number of items sold (x)
:
They tell you two things
Price (p) = 2x^2 - 100
and
The number of items is given as the value of x
:
We know that revenue (R) = number of items sold (x) * the price (p), therefore:
R = x * (2x^2 - 100)
:
Multiply what's inside the brackets and you have:
R = 2x^3 - 100x; this is what they want
:
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Simplify: the square root of 48 x squared y to the 6th
:

:
Factor this inside the radical to recognize the perfect squares:

:
Extract the square roots and you have:

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A projectile is thrown upward at the rate of 42 feet per second from a platform 150 feet high.
:
How long does it take to hit ground?
Let t = time in seconds
;
The three parts of this equation are:
Gravity (downward): -16t^2
Velocity up-ward: 42t
Platform ht: 150
:
The height of the ground is 0, find t when the equation = 0:
-16t^2 + 42t + 150 = 0
:
Unfortunately, this has to be solved by using the quadratic formula:
a = -16; b = 42; c = 150, find t:

:

:

:

:

:

Two solutions:


t = -2.01, we don't want the negative solution
and


t = +4.64375 seconds to hit the ground
:
:
You can check this by substituting 4.64375 for t in the original equation

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Use substitution to solve:
x = 3y - 2
2x + 4y = 16.
Substitute (3y-2) for x in the second equation, solve for y
2(3y-2) + 4y = 16
6y - 4 + 4y = 16
6y + 4y = 16 + 4
10y = 20
y = 20/10
y = 2
Find x substitute 2 for y
x = 3(2) - 2
x = 6 - 2
x = 4
:
:
Use substitution to solve:
2x + y = 7
y = (7 - 2x); we can substitute (7-2x) for y in the next equation, find x
5x - 2y = 4
5x - 2(7-2x) = 4
5x - 14 + 4x = 4; remember a neg outside the brackets changes the signs inside
5x + 4x = 4 + 14
9x = 18
x = 18/2
x = 2
Find y using y = (7-2x)
y = 7 - 2(2)
y = 7 - 4
y = 3
:
:
Find the solution of the system,
y = x^2 + 5
y = 2x + 4
Substitute (x^2 + 5) from the 1st equation, for y in the 2nd equation:
x^2 + 5 = 2x + 4
x^2 - 2x + 5 - 4 = 0
x^2 - 2x + 1; a quadratic equation that we can factor to:
(x-1)(x-1)
x = 1
Find y using the 2nd equation:
y = 2(1) + 4
y = 2 + 4
y = 6
Check it by substitution in the 1st equation
:
:
Solve the system of equations by graphing:
y = x^2 - 4
y = x - 2
Plot these. substitute for x = -3 to x = +3
The first equation table
x | y
-------
-3 | +5
-2 | 0
-1 | -3
0 | -4
+1 | -3
+2 | 0
+3 | +5
Note that this is a parabola
:
The 2nd equation is linear, two points are sufficient
x | y
-------
-3 |-5
+3 |+1
:
The graph should look like this:

:
There are two solutions, where the two graphs intersect. I'll let you figure
out what they are from the graph. When you have decided what they are, check
by substitution in the original equations.

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radiator holds 5 gallons. The coolant in the radiator is a mixture of antifreeze and water. Presently, 40% of the mixture is pure antifreeze. The proper mixture to prevent freezing and rust formation is when antifreeze is 50% of the mixture. The radiator is now full to the top. In order to add antifreeze, some of the present mixture must be drained out to make room. How much mixture should be drained out to make just enough room to add the correct amount of pure antifreeze to give a resulting mixture in the radiator of 50%?
:
Let x = amt to be removed; & amt of pure antifreeze to be added
:
.40(5-x) + 1.0x = .50(5)
:
2 - .4x + 1x = 2.5
:
.6x = 2.5 - 2
:
.6x = .5
:
x = .5/.6
:
x = gal removed, and gal of pure antifreeze added:
:
:
Check solution using decimals = .833:
:
5 - .833 = 4.167 gal of 40% solution
:
.4(4.167) + 1.0(.833) = .5(5)
1.67 + .833 = 2.50, confirms our solution

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The denominator of a fraction is one more than the numerator. When 4 is added to the numerator and 6 is added to the denominator the result equals 3/4. Find the fraction.
this is what I have: , this is right
:
Right here you should have:
=
:
=
:
Cross multiply and you have:
4(x+4) = 3(x+7)
:
4x + 16 = 3x + 21
4x - 3x = 21 - 16
:
x = 5
:
:
Checking our solution using the original statement:
=
=

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Think of a real-life situation that can be translated to a polynomial function and solve the problem.
:
you have to earn at least $3000 this summer, to be able to return to college:
:
You found a construction job that will let you take home $120 per day.
:
The job site is 50 miles from your home. You must drive 100 mi every day.
:
The car you own costs 25 cents a mile to operate (It's an old one).
:
You bring your own lunch (using your parents food) so no expense there.
:
How many days will you have to work to have $3000 at the end of the summer.
:
Let x = number of days worked, f(x) = 3000
:
f(x) = 120x - x(.25*100)
:
Wages - car expense = required amt
120x - x(.25*100) = 3000
120x - x(25) = 3000
120x - 25x = 3000
95x = 3000
x = 3000/95
x = 31.58 days, so let's say you must work 32 days to reach your goal goal
:
:
Check it using 32 days
120(32) - 32(25) =
3840 - 800 = $3040.00
:
Actually this is a real life situation that occurred not too long ago

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x^2 - x/ x^2 + 3x + 2 / x^2-1/x^2+2X
:
Assume you mean:

-------------

:
Can be factored to:

-------------

:
Invert the dividing fraction and multiply:
*
:
CAncel the (x-1)'s and (x+2)'s and you have:
* =

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- = 1
:
You can see that (x^2-1) is the difference of squares so factor that:
- = 1
:
Common denominator would be (x-1)(x+1), multiply equation by that:
(x-1)(x+1) - (x-1)(x+1) = 1(x-1)(x+1)
:
Cancel out the denominators and you have;
3(x+1) - 6 = (x-1)(x+1)
:
Multiply what's inside the brackets on the left, FOIL the right
3x + 3 - 6 = x^2 - 1
3x - 3 = x^2 - 1
:
Arrange as a quadratic equation:
x^2 - 3x - 1 + 3 = 0
x^2 - 3x + 2 = 0
:
Factors to:
(x-2)(x-1) = 0
Two solutions:
x = +2
x = +1
:
However x = 1, cannot be a solution. Note that if you substitute 1 for x,
both fractions have division by 0
:
x = 2 is the only solution, check by subsitution
:
- = 1
3 - = 1
:

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If two inlet pipes are both open, they can fill a pool in 1 hour and 12 minutes. one of the pipes can fill the pool by itself in 2 hours. How long would it take the other pipe to fill the pool by itself
:
Change 1 hr 12 min to hrs. 12/60 = .2 > 1.2 hrs
:
Let x = time for other pipe to fill by itself
:
Let the full pool = 1
:
+ = 1
:
Multiply equation by 2x:
2x* + 2x* = 2x*1
;
Cancel out the denominators and you have
1.2x + 1.2(2) = 2x
:
2.4 = 2x - 1.2x
:
2.4 = .8x
:
x = 2.4/.8
:
x = 3 hrs required by the other pipe alone
:
:
Check solution:
+ =
.6 + .4 = 1

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Simplify:
:
3x + 2n = (4y-2+3x)
:
3x + 2n = 4y - 2 + 3x
:
Subtract 3x from both sides:
2n = 4y - 2
:
Divide equation by 2 and you have:
n = 2y - 1

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a steamer going downstream in a river covers the distance between two towns in 20 hour coming back upstream it covers this distance in 25 hours the speed of water is 4 km/h. find the distance between the two town.
:
Let s = speed of the boat in still water
then
(s-4) = speed upstream
and
(s+4) = speed downstream
;
Distance both ways is equal. Make a distance equation; Distance = Time * speed
Up distance = down distance
25(s-4)= 20(s+4)
:
25s - 100 = 20s + 80
25s - 20s = +100 + 80
5s = 180
s = 180/5
s = 36 mph
:
Find the distance:
Dist = time * speed
downstream: d = 20(36+4) = 800 mi
and
Upstream: d = 25(36-4) = 800 mi also

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Every time I try to solve this problem, nothing comes out the same and I can't seem to figure out where my mistake is to determine if any of my solutions are correct. Can someone please show me the correct way to solve this problem?
. Divide:
Assume you mean

------------

:
Let's do the numerator fraction and the denominator fraction separately:
:
Simplify the numerator fraction:

:
Note that you can factor both denominator and numerator to:

:
Note that the (x-1)'s will cancel leaving you with:

:
:
Simplify the denominator fraction

:
Note that you can factor out the 7's

:
7's cancel, leaving you with:

:
Put it all back together and you have:

---------

:
Remember you invert the dividing fraction and multiply:
* = 1
:
Note that everything cancels
:
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solve : 3 less than the quotient of a number and 7 is 6.
:
Just write what it says, (the word "is" usually means =)
:
Let x = "a number"
:
The quotient of a number and 7 =
:
The equation:
- 3 = 6
:
Add 3 to both sides:
= 6 + 3
:
= 9
:
Multiply equation by 7
7* = 7*9
:
Cancel the 7's and you have:
x = 63
:
:
Check solution in original equation:
- 3 = 6
9 - 3 = 6

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Suppose that Wendy rides her bicycle 30 miles in the same time that it takes Kim to ride her bicycle 20 miles. IF Wendy rides 5 miles per hour faster than Kim, find the rate of each.
:
Let s = K's speed
then
(s+5) = W's speed
:
Since they tell us the the two times are the same, we can make a simple time
equation from this fact: Time = Distance/Speed
:
K's time = W's time
=
:
Cross multiply
30s = 20(s+5)
:
30s = 20s + 100
:
30s - 20s = 100
:
10s = 100
:
s = 10 mph is K's speed and 15 mph is W's speed
:
:
Check solution by find the time of each:
20/10 = 30/15
2 = 2

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Ryan can mow a lawn in 1 hour, and his son, Malik, can mow the same lawn in 50 minutes. One day Malik started mowing the lawn by himself, and worked for 30 minutes. Then Ryan joined him, and they finished the lawn. How long did it take them to finish mowing the lawn, after Ryan started to help?
:
Assuming they have two lawn mowers.
:
Let x = time required to finish the lawn, together
Let the completed job = 1
:
:
M, by himself + the two together = completed job
+ + = 1
:
Multiply equation by 300 to get rid of the denominators:
300* + 300* + 300* = 300*1
:
Cancel out the denominators and you have:
6(30) + 6x + 5x = 300
:
180 + 11x = 300
:
11x = 300 - 180
:
x = 120/11
:
x = 10.91 minutes
:
:
Check solution on a calc:
+ + =
.6 + .2182 + .1818 = 1.0000

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At exactly what time (to the fraction of a minute) between 7:00 and 8:00 do the hands of a clock form a straight angle?
:
Let x = no. of minutes
:
Convert minutes to degrees, 360/60 = 6 deg/min
:
Convert hours to degrees, 360/12 = 30 deg/hr
:
7 o'clock = 7 * 30 = 210 degrees
:
Portion of an hr = (x/60) * 30 = x/2 or .5x
:
Hr hand degrees - minute hand degrees = 180
210 + .5x - 6x = 180
:
-5.5x = 180 - 210
:
-5.5x = -30
:
x = -30/-5.5
:
x = 5.45 minutes. The time would be 07:05.45 for a straight line of the hands
:
Check our solution
210 + (5.45/60)30 - 5.45(6) =
210 + 2.7 - 32.7 = 180

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A small firm produces both AM and AM/FM car radios. The AM radios take 15 h to produce, and the AM/FM radios take 20 h. The number of production hours is limited to 300 h per week. The plant's capacity is limited to a total of 18 radios per week, and existing orders require that at least 4 AM radios and at least 3 AM/FM radios be produced per week. Write a system of inequalities representing this situation. Then, draw a graph of the feasible region given these conditions, in which X is the number of AM radios and Y is the number of AM/FM radios.
:
Let x = number of AM radios; let y = number of AM/FM radios

The production hour constraint:
15x + 20y =< 300
Arrange in general form to plot the graph
20y =< 300 - 15x
y =< 300/20 - (15/20)x
y =< 15 - .75x, use this to plot the production hr graph (Purple line)
:
Plant's capacity constraint:
x + y =< 18
y =< 18 - x, plot this graph also (Green line)
:
Min AM radio production constraint:
x => 4
This will be a vertical line going thru the x axis at +4, (I can't draw this in)
:
Min AM/FM radio production constaint
y => 3
This will be a horizontal line going thru the y axis at +3, (Dark blue line)
:
Here is the graph as presented by these equations except for x => 4,
:

:
The feasibility region:
1. At or below the purple or green line, whichever is lowest
2. At or above the horizontal line
3. At or to the right of the vertical line which you have to draw in at x = +4
:

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A rectangular garden is to be surrounded by a walkway of constant width. The garden's dimensions are 30ft by 40ft. The total area, garden plus walkway, is to be 1800ft^2. What must be the width of the walkway to the nearest thousandth?
:
Let x = width of the walkway
:
Draw this out labeling the walkway width as x, and the inner rectangle
representing the garden as 30 by 40. It will be apparent that the overall
dimensions will be (2x+30) by (2x+40), the area of that is given as 1800 sqft
:
(2x + 30) * (2x + 40) = 1800
:
FOIL
4x^2 + 140x + 1200 = 1800
:
4x^2 + 140x + 1200 - 1800 = 0
:
4x^2 + 140x - 600 = 0
:
Simplify, divide eq by 4 and you have:
x^2 + 35x - 150 = 0
:
Use the quadratic formula to find x: a=1; b=35; c=-150

:

:

:
Do the math and you will get two solutions; -38.86 and +3.86,
obviously it's x = 3.86
:
3.86 ft is the width of the walkway
:
:
Check it:
2x = 2(3.86) = 7.72
So we have:
37.72 * 47.72 = 1799.998 ~ 1800 sq ft
:
How about this? Did it make sense to you? Any questions?

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At a well-known university, 1/4 of the undergraduates commute; 1/3 of the graduates commute. 1/10 of the undergraduates drive more than 40 miles daily and 1/6 of the graduates drive more than 40 miles daily. If there are twice as many undergrads as there are grad students, then what fraction of the commuters drive more than 40 miles daily?
:
Let's use a convenient number for the total students: 720 has a lot of factors
:
It says,"there are twice as many undergrads as there are grad students,"
That would give us us 480 undergrads, and 240 grads
:
1/4 of the undergraduates commute;
* 480 = 120
:
1/3 of the graduates commute.
* 240 = 80
:
1/10 of the undergraduates drive more than 40 miles daily
* 480 = 48
:
1/6 of the graduates drive more than 40 miles daily.
* 240 = 40
:
what fraction of the commuters drive more than 40 miles daily?
Total commuters: 120 + 80 = 200
Total commuters drive more than 40 mi: 48 + 40 = 88
:
Then: = of the commuters commute over 40 mi
:
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Let's take it step-by-step
:
"There is a rectangle whose perimeter is 18 centimeters."
Let the length and width be x & y
Perimeter:
2x + 2y = 18
Simplify, divide equation by 2
x + y = 9
y = (9-x); we substitute (9-x) for y
Find the area:
x(9-x) = -x^2 + 9x
:
"If its length is decreased by 5 centimeters and it's width is increased by 12 centimeters, it's area is doubled. Find its length and width."
:
New length: (x-5)
:
New width:
(9-x) + 12
(-x + 21)
:
New area (double the old area): 2(-x^2 + 9x) = -2x^2 + 18x
:
New width * new length = twice the area
(-x + 21)*(x - 5) = -2x^2 + 18x
FOIL the left side
-x^2 + 5x + 21x -105 = -2x^2 + 18
-x^2 + 26x - 105 = -2x^2 + 18x
:
Combine to form a quadratic equation:
-x^2 + 2x^2 + 26x - 18x - 105 = 0
x^2 + 8x - 105 = 0
Factor this to
(x + 15)*(x - 7) = 0
x = +7 is the original length (only the positive solution is used)
:
Remember y = 9 - x:
y = 9 - 7
y = 2 is the original width
Original area = 7 * 2 = 14 sq/cm
:
New length and width:
L = 7 - 5 = 2 cm
W = 21 - 7 = 14
2 * 14 = 28 sq/cm, double the original
:
Could you follow this OK?

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Dorothy drives 10 kilometers, then increases her speed by 10 kilometers per hour and drives another 25 kilometers. Find her original speed if she drove for 45 minutes.
:
Let s = original speed
then
Increased speed = (s+10)
:
Write a time equation: Time = distance/speed; Change 45 min to hrs: 45/60 = .75 hr
:
Time for 10 km + time for 25 km = .75 hr
+ = .75
:
Multiply equation by s(s+10) and you have:
10(s+10) + 25s = .75(s(s+10)
:
10s + 100 + 25s = .75s^2 + 7.5s
:
35s + 100 = .75s^2 + 7.5s
:
Arrange as a quadratic equation:
.75s^2 + 7.5s - 35s - 100 = 0
:
.75s^2 - 27.5s - 100 = 0
:
Use the quadratic formula to solve for s: a=.75; b=-27.5; c=-100

:

:

:

:

:
; we only want the positive solution here
:

:
s = 40 mph is her speed for the 1st 10 km (original speed
:
:
Check our solution using the time equation:
+ = .75 as given
:
:
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Two planes are 6,000 miles apart, and their speeds differ by 200 mph. They travel toward each other and meet in 5 hours. Find the speed of the slower plane.
:
Let t = speed of the slower plane
then
(t+200) = speed of the faster plane
:
The travel time of the planes are the same; (5 hrs)
:
Write a distance equation: Distance = time * speed
:
Fast plane dist + slow speed dist = 6000
5(s+200) + 5s = 6000
5s + 1000 + 5s = 6000
10s = 6000 - 1000
10s = 5000
s = 5000/10
s = 500 mph is the slow plane
:
Check solution
Fast plane: 500 + 200 = 700 mph
5*700 + 5*500 =
3500 + 2500 = 6000 miles

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A red candle is 45 cm tall and burns 3 cm per hour, a white candle is 30 cm tall and burns 1-1/2 cm per hour, how tall will each candle be after 1 hour, 3 hours, 5 hours and 12 hours?
:
What is the formula for calculating each candle's height at each time
Let t = time in hrs
:
Red: Candle ht = 45 - 3t
White: Candle ht = 30 - 1.5t
:
Substitute 1, 3, 5 and 12 in each equation
:
Example: using the Red candle burning 12 hrs
ht = 45 - 3(12)
ht = 45 - 36
ht = 9 cm
:
:
which candle lasts longer?
Red
45 - 3t = 0
-3t = -45
t = -45/-3
t = +15 hrs duration of the Red candle
:
White
30 - 1.5t = 0
-1.5t = -30
t = -30/-1.5
t = 20 hrs duration of the white candle
:
:
Will the red and white candles ever be the same height at the same time if started at the same time?
:
White ht = Red ht
30 - 1.5t = 45 - 3t
3t - 1.5t = 45 - 30
1.5t = 15
t = 15/1.5
t = 10 hrs they will be equal
:
You can determine that height that is, substitute 10 for t in either equation
and solve for ht

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I am having major problems with this
(x+5)^2 = 3
Find the square root of both sides:
x + 5 = +/-Sqrt(3)
Two solutions:
x = -5 + Sqrt(3)
and
x = -5 - Sqrt(3)

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how many ounces of pure water must be added to 75 ounce of a 20% salt solution to make a salt solution which is 15% salt?
:
Let x = amt of water to be added:
:
.20(75) = .15(75+x)
:
15 = 11.25 + .15x
:
15 - 11.25 = .15x
:
3.75 = .15x
:
x = 3.75/.15
:
x = 25 ounces of pure water to be added
:
:
Check solution using the salt content:
.20(75) = .15(75 + 25)
15 = 15

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A company produces both large and small cabinets. A small cabinet requires 1 hour of labor and a large cabinet requires 4 hours of labor. The company has at most 80 hours of labor available each day. No more then 60 small cabinets and no more than 15 large cabinets can be produced in a day due to space limitations. If the company's profit is $120 per small cabinet and $250 per large cabinet, how many of each should be produced to maximize profit? What is the maximum profit?
:
Let x = no. of small cabinets
Let y = no. of large cabinets
:
Write an equation for each constraint:
:
The labor hrs equation
x + 4y =< 80
or
4y =< 80 - x
y =< 80/4 - x/4
y =< 20 - .25x; use this for graphing
:
The Production space equation:
x =< 60; no more that 60 small cabinets
y =< 15: no more that 15 large cabinets
:
Graph all these equations:
Here is the graph, you have to complete the graph by drawing a vertical line
thru x = 60

:
The area of feasibility will be:
1.At or to the left of the vertical line that you draw in
2.At or below the horizontal line (purple) OR
3.At of below the green line, which ever is lowest.
:
You can calculate the corner coordinate values which bound the feasibility area
Use this to find the profit
Use x = 60 and y = 20-.25x to find that corner:
Substitute x = 60 in y = 20=.25x
y = 20 - .25(60)
y = 20 - 15
y = 5
:
Objective function for x=60, y=5
60(120) + 5(250) =
7200 + 1250 = $8450 profit, when you sell 60 small cab and 5 lg cab
:
:
Use y = 15 to find x for the other corner
15 = 20 - .25x
.25x = 20 - 15
x = 5/.25
x = 20
:
Objective function for x = 20, y = 15
20(120) + 15(250) =
2400 + 3750 = $6150 profit, when you sell 20 small cab and 15 lg cab
:
The 3rd corner is x = 0, y = 15, obviously would not be max profit
:
Did this make sense to you?

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Let x = the 10's digit, and y = the units digit
Write an equation for each statememt
:
"sum of the digit of two digit no. is 12."
x + y = 12
or
y = 12 - x; we use this for substitution, if needed
;
"the given no. exceed the no. obtained by interchange the digit by 36."
:
Given number: 10x + y
interchanged number: 10y + x
:
The equation from the given statement:
10x + y = 10y + x + 36
Some simple algebra:
10x - x = 10y - y + 36
9x = 9y + 36
Simplify, divide equation by 9
x = y + 4
:
Substitute (12-x) for y, (from the 1st statement)
x = (12-x) + 4
x + x = 12 + 4
2x = 16
x = 8
:
y = 12 - 8 = 4
:
Our number is 84:
:
:
Check it: 84 - 48 = 36

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The sum of a number and it's reciprocal is 73/24. find the number.
Please help.
:
x + =
:
Multiply equation by 24x and you have:
24x^2 + 24 = 73x
:
24x^2 - 73x + 24 = 0; our old friend, the quadratic equation
:
After some tedious figuring got it to factor to:
(8x - 3)(3x -8) = 0
8x = 3
x =
and
3x = 8
x =
:
Check solution using 8/3:
+ =
+ =
:
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car travels 15 kilometers per hour faster than a truck. The car goes 360 kilometers in 2 hours less time than it takes the truck to travel the same distance. Find the speed of the truck and the car.
:
Let s = speed of the truck
then
(s + 15) = speed of the car
:
Write a time equation: Time = Distance/speed
:
Truck time - car time = 2 hrs
- = 2
:
Multiply equation by s(s+15) to get rid of the denominator, results:
360(s+15) - 360s = 2(s(s+15)
:
360s + 5400 - 360s = 2(s^2 + 15s)
:
5400 = 2(s^2 + 15s)
:
Simplify, divide both sides by 2
2700 = s^2 + 15s
:
s^2 + 15s - 2700 = 0; a quadratic equation
:
Factors to:
(s + 60) (s - 45) = 0
:
s = +45 mph is the truck (positive solution)
:
45 + 15 = 60 mph is the car
:
:
Check solution by finding the time of each:
360/45 = 8 hrs
360/60 = 6 hrs
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difers = 2 hrs as given in the problems

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