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-x+2y=-10
5x+4y=8
.
Multiply top equation by -2 and combine with second:
2x-4y = 20
5x+4y = 8
--------------
7x = 28
x = 4
.
now, substitute the value found above into either of the two equations to find y:
5x+4y=8
5(4)+4y=8
20+4y=8
4y=-12
y = -3
.
answer: x=4 and y=-3

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Alex can complete a work in 5 hours and Ben can complete the same work in 10 hours .How many hours will they both take to complete the job together?
.
Let x = time (hours) for both
then
x(1/5 + 1/10) = 1
multiplying both sides by 10:
x(2 + 1) = 10
x(3) = 10
x = 10/3
x = 3 and 1/3 hours
or
x = 3 hours and 20 minutes

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Find X so that log base 3 of X + log base 3 of (X-3) = log base 3 of 10.
.






x = {-2, 5}
we can throw out the -2 (extraneous -- won't work) leaving:
x = 5

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A container has 6 litres f a 40% alcohol solution in it. How much pure alcohol should be added to raise it to be a 60% solution?
.
Let x = amount (liters) of pure alcohol added
then
x + .40(6) = .60(x+6)
x + 2.4 = .60x+3.6
.40x + 2.4 = 3.6
.40x = 1.2
x = 3 litres

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Find a polynomial function of lowest degree with integer coefficients that has the given zeros.
5, −6, 2
.
If you have the above zeros, then the factors must be:
(x-5)(x+6)(x-2)
multiplying them together gives the polynomial.
(x-5)(x+6)(x-2)
(x^2+x-30)(x-2)
x^2(x-2) + x(x-2) - 30(x-2)
(x^3-2x^2) + (x^2-2x) - (30x-60)
x^3-2x^2 + x^2-2x - 30x+60
x^3-x^2 -2x - 30x+60
x^3-x^2-32x+60

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Find a polynomial f(x) of degree 3 with real coefficients and the following zeros: -2, 3+i
.
If a zero has an "imaginary" root such as:
3+i
then, there must be another imaginary root that is the conjugate:
3-i
.
so, all the zeros are:
-2, 3+i , 3-i
.
The factors must be:
(x - (-2)) , (x - (3+i)) , (x - (3-i))
(x+2) , (x-3-i) , (x-3+i)
.
Multiply the factors together to get the polynomial:
(x+2)(x-3-i)(x-3+i)
(x+2)(x(x-3+i)-3(x-3+i)-i(x-3+i))
(x+2)((x^2-3x+xi)-(3x-9+3i)-(xi-3i+i^2))
(x+2)((x^2-3x+xi)-(3x-9+3i)-(xi-3i-1))
(x+2)(x^2-3x+xi-3x+9-3i-xi+3i+1)
(x+2)(x^2-6x+xi+9-3i-xi+3i+1)
(x+2)(x^2-6x+9-3i+3i+1)
(x+2)(x^2-6x+9+1)
(x+2)(x^2-6x+10)
x(x^2-6x+10)+2(x^2-6x+10)
(x^3-6x^2+10x)+(2x^2-12x+20)
x^3-6x^2+10x+2x^2-12x+20
x^3-4x^2+10x-12x+20
x^3-4x^2-2x+20

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to solve, you can apply the "quadratic formula":
[-b +- sqrt(b^2 - 4ac)]/(2a)
[-(-7) +- sqrt((-7)^2 - 4(2)(-5))]/(2*2)
[7 +- sqrt(49 - (-40))]/(4)
[7 +- sqrt(49+40)]/4
[7 +- sqrt(89)]/4
but, since 89 is "prime" you cannot simplify underneath the radical.
giving you the answer of:
7/4 +- sqrt(89)/4

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how many liters of a 10 alcohol solution must be mixed with 80 liters of a 70% solution to get a 30% solution?
.
Let x = amount (liters) of 10% alcohol added
then
.10x + .70(80) = .30(x+80)
.10x + 56 = .30x+24
56 = .20x+24
32 = .20x
32/.20 = x
160 liters = x

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How do you write a quadratic equation in the form ax^2 + bx + c = 0 such that c = 4 and the equation has the given solutions of -4 and 3?
because you are given that c=4 we have two equations and two unknowns:
since x= -4 is a solution:
a(-4)^2 + b(-4) + 4 = 0
16a - 4b + 4 = 0 (equation 1)
.
since x= 3 is a solution:
a(3)^2 + b(3) + 4 = 0
9a + 3b + 4 = 0 (equation 2)
.
System of equation:
16a - 4b + 4 = 0
9a + 3b + 4 = 0
.
multiply top by 3 and bottom by 4:
48a - 12b + 12 = 0
36a + 12b + 16 = 0
-----------------------
84a + 28 = 0
84a = -28
a = -28/84
a = -1/3
.
plug above into:
9a + 3b + 4 = 0
9(-1/3) + 3b + 4 = 0
-3 + 3b + 4 = 0
3b + 1 = 0
3b = -1
b = -1/3
.
our final answer:
(-1/3)x^2 - (1/3)x + 4 = 0

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The amount of water in the leaking tant is inversely proportional to the time is has been leaking. If after 4 hours there are 50 litters left in the tant. How many litters will be left after 5 hours?
"inversely proportional" think:
y = k/x
to find 'k', use "after 4 hours there are 50 litters left in the tank"
50 = k/4
200 = k
.
Our relationship then, is
y = 200/x
Now, we can answer:
How many litters will be left after 5 hours?
y = 200/5
y = 40 liters

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a car travelled a distance of 89 km in 6 hours.What is it's speed
.
That's:
89 km per 6 hours
or
89/6
14.83 km per hour (answer)

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8x^3y^9
can be written as:
2^3x^3(y^3)^3
2xy^3 (answer)

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how far could you walk in 20 seconds ?
.
Assuming a person can walk 3 mph
then
3 mph * 20 secs * 1/60 min/sec * 1/60 hr/min
3 * 20 * 1/60 * 1/60
60 * 1/60 * 1/60
1 * 1/60
1/60 mile
or
1/60 mile * 5290 ft/mile = 88 feet

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given f(x)= (7/x)-3 find f^-1(x)
.
f(x)= (7/x)-3
y= (7/x)-3
switch x for y and y for x:
x= (7/y)-3
solve for y:
xy = 7-3y
xy+3y = 7
y(x+3) = 7
y = 7/(x+3)
.
so:
f^-1(x) = 7/(x+3)

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A man has 26 coins in his pocket, all of which are dimes and quarters. If the total value of his change is 485 cents, how many dimes and quarters does he have?
.
Let x = number of dimes
then
26-x = number of quarters
.
10x + 25(26-x) = 485
10x + 650-25x = 485
650-15x = 485
-15x = -165
x = -165/(-15)
x = 11 (dimes)
.
quarters:
25-11 = 14

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Suppose that a 12 quart radiator contains a 20% solution of antifreeze. how much solution needs to be drained out and replaced with pure antifreeze to obtain a 40% solution of antifreeze
.
Let x = amount (quarts) drained and replaced
then
.20(12-x) + x = .40(12)
2.4-.20x + x = 4.8
2.4 + .80x = 4.8
.80x = 2.4
x = 2.4/.80
x = 3 quarts

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You have a typo in the problem as stated. Please correct and resubmit.

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The reciprocal of:
9i
is
1/(9i)
.
But, we don't want i in the denominator so we multiply by:
1/(9i) * i/i
i/(9i^2)
i/(-9)
or
(-1/9)i

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Chocolate Confections, Inc. has determined that when x chocolate cakes are made daily, the average cost per chocolate cake is given by C(x)=0.001x^2-0.14x+12.20
a) What is the average cost per cake if 30 chocolate cakes are made daily?
set x to 30 and solve for C(x):
C(x)=0.001x^2-0.14x+12.20
C(30)=0.001(30)^2-0.14(30)+12.20
C(30)=0.9-4.2+12.20
C(30)=-3.3+12.20
C(30)=$8.90
.
b) How many chocolate cakes should be made daily in order to minimize the average cost per cake? Show work
The given equation:
C(x)=0.001x^2-0.14x+12.20
is a parabola that opens upwards, therefore the vertex is the minimum:
x = -b/(2a)
x = -(-0.14)/(2*0.001)
x = 0.14/(0.002)
x = 70 cakes

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Find the value of x in the equation 14^x = 0.4285.
.
14^x = 0.4285
x =
x =
x = -0.321

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20%of14
20/100 * 14
.20 * 14
2.8

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One number is one-third of another. The difference of their squares is ¾. Find the numbers
.
Let x = one number
then
x/3 = other number
.
x^2 - (x/3)^2 = 3/4
x^2 - x^2/9 = 3/4
multiplying both sides by 36:
36x^2 - 4x^2 = 27
32x^2 = 27
x^2 = 27/32
x =
x =
.
other number:
x/3 =
x/3 =

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Find the length and width of a rectangle if its length is thrice its width and its diagonal is 7.
.
Let w = width
then
3w = length
.
applying Pythagorean theorem:
w^2 + (3w)^2 = 7^2
w^2 + 9w^2 = 49
10w^2 = 49
w^2 = 4.9
w = 2.2 (width)
.
length:
3w = 3(2.21) = 6.6

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Using two even consecutive integers where two times the lower integer plus the higher integer equals 50.
.
Let x = lower even integer
then
x+2 = higher consecutive even integer
.
2x + (x+2) = 50
3x + 2 = 50
3x = 48
x = 16 (lower)
.
higher:
x+2 = 16+2 = 18
.
answer: 16 and 18

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An airline carries 180 passengers and 10 tonnes of freight and earns $75000 for its owners in fares and freight charges. On the return trip it carries 100 passengers and 20 tonnes of freight, earning $85000. What is the passenger fare per person and the freight charges per tonne.
.
Let x = cost ($) per person
and y = cost ($) per ton
.
from:"180 passengers and 10 tonnes of freight and earns $75000"
180x + 10y = 75000
.
from:"100 passengers and 20 tonnes of freight, earning $85000"
100x + 20y = 85000
.
Our simultaneous equations:
180x + 10y = 75000
100x + 20y = 85000
.
multiply top equation by -2 and combine with second:
-360x - 20y = -150000
100x + 20y = 85000
------------------------
-260x = -65000
x = -65000/(-260)
x = $250 (passenger fare per person)
.
freight charges per tonne:
plug above into:
100x + 20y = 85000
100(250) + 20y = 85000
25000 + 20y = 85000
20y = 60000
y = $3000 (per tonne)

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solve the equation 4s^2 + 36s + 81 = 0
.
4s^2 + 36s + 81 = 0
4s^2 + 18s + 18s + 81 = 0
(4s^2 + 18s) + (18s + 81) = 0
2s(2s + 9) + 9(2s + 9) = 0
(2s + 9)(2s + 9) = 0
(2s + 9)^2 = 0
s = -9/2 = -4.5

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When does cos(x) = 0?
From the "unit circle" cos(x)=0 when
In terms of degrees: 90 and 270 degrees
In terms of radians: (pi)/2 and 3(pi)/2 radians

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A bank teller has $22.55 in quarters and dimes. There are 58 more quarters than dimes. How many dimes does he have?
.
Let d = number of dimes
then because "There are 58 more quarters than dimes" we have
d+58 = number of quarters
.
.10d + .25(d+58) = 22.55
.10d + .25d+14.5 = 22.55
.35d+14.5 = 22.55
.35d = 8.05
d = 23 (number of dimes)
.
number of quarters:
d+58 = 23+58 = 81

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a man can complete a piece of work in 9 days but with the help of his son, he can do it in 6 days.the taken by his son to do it alone is
.
Let x = time (days) it takes the son alone
then
6(1/9 + 1/x) = 1
multiplying both sides 9x:
6(x + 9) = 9x
6x + 54 = 9x
54 = 3x
18 days = x

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how many questions can I get wrong on a 36 question test in order to achieve a score of 68% with a max score of 100?
Let x = number of questions got wrong
then
(36-x)/36 = .68
multiplying both sides by 36:
36-x = 24.48
36 = 24.48+x
11.52 = x
You can miss 11 questions

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Dan earned money from babysitting. He spent 1/4 of his money on a guitar and gave 1/4 of what was left to charity. He has $108 left. How much money did he begin with?
Let x = amount ($) he began with
then
x - (1/4)x - (1/4)(x - (1/4)x) = 108
x - x/4 - ((1/4)x - (1/4)(1/4)x) = 108
x - x/4 - (x/4 - (1/16)x) = 108
x - x/4 - (x/4 - x/16) = 108
x - x/4 - x/4 + x/16 = 108
multiplying both sides by 16:
16x - 4x - 4x + x = 1728
16x - 8x + x = 1728
8x + x = 1728
9x = 1728
x = $192