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x = donut revenue
y = muffin revenue
x + y = 44000
so
y = 44000 - x
.
x = 4.5*y
.
substituting
x = 4.5(44000-x)
multiply though
x = 198000 - 4.5x
add 4.5x to both sides
5.5x = 198000
divide both sides by 5.5
x = 36000
.
y = 44000 - 36000 = 8000
.
Answer:
donut revenue = $36,000
muffin revenue = $8,000

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x^2 - 18x + 18 = 0

Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=252 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 16.9372539331938, 1.06274606680623. Here's your graph:

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x=8y-10
x=4y-5
.
x - 8y = -10
x - 4y = -5
subtract
-4y = -5
y = 5/4
.
substituting
x = 8y -10
x = 8(5/4) - 10 = 0
.
Solution: (0, 5/4)
.
Check by inspecting the graphs:

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Solution problems are solved most easily by keeping track of the amount of 'pure' stuff in the constituent solutions and in the final solution.
.
Here you want 12 liters of a 6% solution of silver iodide.
That means you need .06*12 = .72 liters of 'pure' silver iodide.
.
You have a 4% and a 10% solution to add together.
'x' can be used to indicate the amount of 4% solution.
We could say 'y' is the amount of 10% solution, but it is likely easier to indicate '12-x' as the amount of 10% solution.
.
We can now setup the equation to be solved:
.04x + .10y = .72
or
.04x + .10(12-x) = .72
.04x + 1.2 - .1x = .72
-.06x = -.48
x = 8
.
y = 12-8 = 4
.
So, our candidates are:
x = 8 liters of 4% solution
y = 4 liters of 10% solution
.
But we have to check to know if that's right...always...
.
How much silver iodide is there in 8 liters of 4% solution?
.04*8 = .32 liters
How much silver iodide is there in 4 liters of 10% solution?
.1*4 = .4 liters
Total = .72 liters of silver iodide
.
That matches what we determined we needed initially.
.
Checking the question, we can see the answer is only "how many liters of the 10% solution"? 4 liters

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The 25-ft wire is cut into two pieces. We can call these 'x' and 'y'. And in algebra, we'd say:
x + y = 25
or
y = 25-x
.
We also are told there is a relationship between the lengths of 'x' and 'y'.
x = 5y + 1 :: 'x' is 1 foot longer than 5 times 'y'
.
substituting y = 25-x, we have
x = 5(25-x) + 1
.
multiplying through
x = 125 - 5x + 1
.
collecting terms
6x = 126
.
dividing by 6
x = 126/6 = 21
.
Since x+y = 25, then
y = 25 - 21 = 4
.
Checking our work
.
Do they sum to 25?
x + y = ??
21 + 4 = 25
Yes
.
Is x = 5*y + 1?
5(4) + 1 = 21 ??
Yes
.
So our answer is:
x = 21 feet
y = 4 feet
.
Done

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When we present numbers we usually do not think about the deeper meaning of what they represent. We simply 'know' that 23 is twenty-three. We think the symbol '23' is a unitary thing. But, in fact, the numbers have place values such that we can say '23' means 2*10 + 3*1. So, the '2' and '3' are simply standing next to one another, they are not multiplied.
.
Now let's think of the number 'xy'. As a student of algebra, you doubtless think of this as x times y.
That is exactly the thinking that will trip you up with number problems like these. Continuing the above example, we could say 'xy' = 23, which means 'x' is 2*10 and y = 3*1. The letters 'x' and 'y' are simply standing beside one another.
.
So we are told there is a positive integer that has two digits. We can call it 'xy'.
And we are told that the value of 'xy' is exactly twice the sum of digits.
.
Well, the sum of digits in the number 'xy' is just:
x + y
.
The value of 'xy' is
10x + y
because the 'x' represents the tens and the 'y' represents the ones.
.
And we are given the equation, namely the value of 'xy' is exactly twice the sum of digis.
10x+y = 2(x+y)
.
Multiplying through
10x + y = 2x + 2y
.
Subtracting y from both sides
10x = 2x + y
.
Subtracting 2x from both sides
8x = y
.
Now where do we go?
.
Well, we can retreat to the logic of numbers that we know.
If 'x' were to be any integer greater than 1, then 'y' would be two digits.
But 'y' cannot be two digits because, by definition, we are told it is a single digit.
Therefore, we logically conclude
x = 1
And since we have shown
8x = y
then we have to conclude
y = 8
.
Thus 'xy' has to be 18, which means the value is 1*10 + 8*1 = 18.
(Remember, they're just standing beside each other, we're not multiplying them.)
.
The sum of digits is 1+8 = 9.
.
Is the value 18 equal to twice the sum of digits? Or mathematically, does
xy = 2(x+y)
.
18 = 2*(1+8) = 2*9 = 18
YES!
.
So the only positive TWO-DIGIT integer that is exactly twice the sum of its digits is 18.
.
Done.

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In general, a quadratic equation may be depicted as y = (x + j)(x + k), where j & k are numbers. The roots of the equation will be the value of x that make x+j = 0 and the value of x that makes x+k = 0.
So, we want the roots to be 2/3 and -1/2, which means:
y = (x-2/3)(x+1/2)
Notice the signs are reversed so the roots will be the value indicated.
Multiplying through, we have
y = x^2 + 1/2x - 2/3x -2/6
y = x^2 + 3/6x - 4/6x - 2/6
y = x^2 - 1/6x - 2/6
.

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For five judge's score to have an average of 9.4, then the total points awarded would have to equal 9.4*5 = 47. This is because the average is the sum of the scored divided by the number of scores.
.
The four judges have awarded: 9.6 + 9.2 + 9.4 + 9.3 points = 37.5.
That means the fifth judge has to award a 9.5 points for the grand total to be 47.

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factoring x^2 + 6x + 5, we arrive at:
y = (x+1)(x+5)
.
This means the intercepts will be -1 and -5
.

.
The vertex is at x= -b/2a = -6/2 = -3.
When x = -3, y = -4, which is the point: (-3,-4).
.
By inspecting the graph, we can see this fits.

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You have two equations that need to put into y=mx+b format.
.
y = x + 13
y = x - 9
.
The slope of both lines is 1, so they may be parallel or the same line.
They have different y-intercepts, so they are not the same line.
So they are parallel.
.

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In solving "solutions" problems, you have to keep track of the amount of pure stuff you need at the end.
.
x = ounces of 20% alcohol solution to add to the 15 oz of 25% solution
.
15 oz of a 25% solution = 3.75 oz of pure alcohol
.
The total amount can be described 15+x ounces.
.
We solve the problem in terms of the amount of pure alcohol
.2x + 3.75 = .23(15+x)
.2x + 3.75 = 3.45 + .23x
.
Subtracting .2x from both sides
3.75 = .03x + 3.45
.
Subtracting 3.45 from both sides
.3 = .03x
Multiply by 100
30 = 3x
Divide by 3
10 = x
x = 10
.
So, you have to add 10 oz of 20% alcohol solution to 15 oz of 25% alcohol.
.
Always check your work.
.
At the end we would have 25 oz that we believe would be 23% alcohol. If that is true, then we would have:
.23 * 25 = 5.75 oz of pure alcohol in the solution.
.
We have shown (above) that we have 3.75 oz of pure alcohol in the 15 oz of 25% solution.
How many oz of pure alcohol is there in 10 oz of 20% alcohol?
.2*10 = 2 oz
.
3.75 + 2 = 5.75 oz, which is exactly what we needed
.
check the question to be sure you answer it at the end...
.
How many ounces do you need to add?
You need to add 10 ounces of 20% alcohol solution.

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x = number of dvd
y = number of cd
x + y = 27
x = 2y - 3
.
set up a simultaneous equations
x + y = 27
x - 2y = -3
subtract
3y = 30
y = 10
.
so
x = 27 - 10 = 17
.
Answer:
10 CDs
17 DVDs
.
Done

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The class has 27 students divided into two groups.
x = number of students in one group
y = number of students in the other group
x + y = 27 describes the whole class.
.
One group has 5 fewer than the other. This can be shown as:
x = y + 5 :: which means y has 5 fewer than x
.
this can be solved as a system of equations.
.
x + y = 27
x - y = 5
add them
2x = 32
x = 16
.
which means
y = 27 - 16 = 11
.
Answer:
One group has 16 and the other has 11.
.
Done.

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With word problems like these, you have to take into consideration that there is one job that will be done in a specific period of time.
.
Tom can do 1 job (the whole job) in 45 minutes. So he can do 60/45 jobs per hr. Maria can do 60/65 jobs per hour.
.
Working together, they can do:
60/45x + 60/65x = 1
(4/3 + 12/13)x = 1
(1.33333 + .92037)x = 1
2.25370x = 1
x = 1/2.2537 = .44371
x = .44371 hr = 26.6226 min
.
You can check to see if their combined work will complete in 26.6226 min.
.
26.6226/45 = .591 of a job
26.6226/65 = .409 of a job
.
Pretty close...

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The radius of a circle is one-half the diameter.

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First get the equations into y = mx+ b format.
.
-3x +4y = 12
4y = 3x + 12
y = 3/4x + 3
.
-6x +8y = 24
8y = 6x + 24
y = 6/8x + 3 = 3/4x + 3
.
so you have only one line
.

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L = length
W = width
P = perimeter = 50
P = 2(L+W) = perimeter formula = 2L + 2W
L - W = 7 :: given
so
L = W+7
.
substituting
2(W+7) + 2W = 50
2W + 14 + 2W = 50
4W = 36
W = 9
.
L=W+7 = 9+7 = 16
.
checking
2L + 2W = 2(16) + 2(9) = 32+18 = 50
.
Answer:
L = 16
W = 9

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Silo A
volume = pi*r^2*30
diameter = 12, so
r = 1/2d = 6
volume = pi*36*30 = 3392.92
it is full to the top, so that is volume of grain.
.
Silo B
has the same amount of grain in it, so it has 3392.92 cubic ft
base diameter = 30
substituting the known volume
3392.92 = pi*15^2*h, where 'h' is the height of the seed in Silo B
h = 3392.92/(pi*15^2)
h = 4.8 ft

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the set {0,1,2,...,170} has 171 elements.
.
of these, there are only 5 perfect cubes are:
1^3 = 1
2^3 = 8
3^3 = 27
4^3 = 64
5^3 = 125
.
so the probability is 5/171

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L = length
W = width
L = W + 5 :: given
P = 2L + 2W :: known definition of the perimeter
P = 34 :: given
.
2L + 2W = 34
2(W+5) + 2W = 34
2W + 10 + 2W = 34
4W + 10 = 34
4W = 24
W = 6
.
L = W+5 = 6+5 = 11
.
checking perimeter
2L + 2W = 2(6) + 2(11) = 12 + 22 = 34
OK.
.
Answer:
L = 11
W = 6
.
Done

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d=rt, where d=distance, r=rate, and t=time
t = d/r= 7
so
x = time walking down @ 4.5 mph
7-x = time walking up @ 2.5 mph
.
total time = 7 = time walking down + time walking up
.
4.5(x) = d = distance walking down
2.5(7-x) = d = distance walking up, which has to equal distance walking down
.
4.5x = 2.5(7-x) = 17.5 - 2.5x
adding 2.5x to both sides
7x = 17.5
x = 2.5 hr
7-x = 7 - 2.5 = 4.5 hr
.
so, she walks down for 2.5 hr at a pace of 4.5 mph = 11.25 miles
and
she walks up for 4.5 hr at a pace of 2.5 = 11.25 miles
.
so the total length of the trip was 11.25 + 11.25 = 22.5 miles
.

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x = price of a case of water
y = price of a case of juice
.
6x = 135 :: 6 cases of water cost $135
.
4y + 2x = 110 :: 4 cases of juice + 2 cases of water cost $110
.
So, there you have 2 equations using x and y.
.
Of course, solving this is now straightforward.
x = 135/6 = 22.50
4y + 2(22.50) = 110
4y = 110 -45 = 65
y = 65/4 = 16.25
.
Done

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By inspection, the first equation, y = -5 is a straight line parallel to the x axis.
That is, for every value of 'x', the value of 'y' is -5.
.
5x + 4y = -20
rearranging
4y = -5x -20
y = -5/4x - 5
which will be a straight line sloping down from the upper left to the lower right, y intercept will be (0, -5); and when x=-4, it will cross the x axis at (-4, 0)
.

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2y = 4x +8
x - 3y = -7
.
rearranging
-4x + 2y = 8
x - 3y = -7
.
multiply the second equation by 4
4x - 12y = -28
.
so, we have
-4x + 2y = 8
4x - 12y = -28
.
add them
-10y = -20
y = 2
.
substitute
2(2) = 4x + 8
4 = 4x + 8
-4 = 4x
x = -1
.
checking
2(2) = 4(-1) + 8
4 = -4 + 8 = 4 TRUE
.
-1 -3(2) = -7 TRUE
.
Answer:
y = 2
x = -1

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x = 1st angle
y = 2nd angle
x = 9y :: given
x + y = 180 :: given
.
substituting for x = 9y
9y + y = 180
10y = 180
y = 18
x = 9y = 9(18) = 162
.
checking...
x + y = 162 + 18 = 180
.
Answer:
x = 162
y = 18
.

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Simplify the

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I assume the problem is to solve:

.
Factoring, we arrive at:

.
Real solutions will be found where

or

.




or

.


There are no real solutions for this part.

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x = longest side of triangle
y = 1/2x :: y is half the longest side
z = x - 10 :: z is 10 less than the longest side
x + y + z = 53
.
substituting
x + 1/2x + x-10 = 53
.
multiply everything by 2
2x + x +2x -20 = 106
.
collect and simplify
5x = 126
x = 126/5 = 25.2
.
y = 1/2x = 25.2/2 = 12.6
.
z = x-10 = 15.2
.
checking, does x+y+z = 53?
25.2 + 12.6 + 15.2 = 53
Yes!
.
Answer:
x = 25.2
y = 12.6
z = 15.2

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3a + 9b = 8b - a
5a - 10b = 4a - 9b + 5
.
for each equation, collect like terms and simplify to get into the right format
.
3a + 9b = 8b - a
add 'a' to both sides
4a + 9b = 8b
subtract 8b from both sides
4a + b = 0
.
5a - 10b = 4a - 9b + 5
subtract 4a from both sides
a - 10b = -9b + 5
add 9b to both sides
a - b = 5
.
now both equations are in the right form
4a + b = 0
a - b = 5
.
add them
5a = 5
a = 1
.
substitute back
.
4a + b = 0
4(1) + b = 0
4 + b = 0
b = -4
.
continue checking...
3a + 9b = 8b - a ??
3(1) + 9(-4) = 8(-4) - 1 ??
3 -36 = -32 -1 ??
-33 = -33 YES!
.
5a - 10b = 4a - 9b + 5 ??
5(1) - 10(-4) = 4(1) - 9(-4) + 5 ??
5 + 40 = 4 + 36 + 5 ??
45 = 45 YES!
.
Answer:
a = 1
b = -4

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T = Tricia's age
L = Lilly's age
E = Emma's age
T = 3L :: Tricia is 3 times as old as Lilly
L = 2E :: Lilly is 2 times as old as Emma
T + L + E = 27 :: given
.
substituting
3L + 2E + E = 27
3L = 3(2E)
6E + 2E + E = 27
9E = 27
E = 3
Emma is 3 years old.
.
L = 2E = 2(3)= 6
Lilly is 6 years old.
.
T = 3L = 3(6) = 18
Tricia is 18 years old.
.
checking...is the total 27?
3+9+18 = 27
.
So the answer is correct.

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x = first positive odd integer
y = second positive odd integer
y = x+2 = the second positive odd integer is consecutive to x.
.
x +y = 130
substituting
x + x+2 = 130
2x + 2 = 130
subtracting 2 from both sides
2x = 128
dividing both sides by 2
x = 64, and it is NOT ODD !
So, there is no solution with odd integers.
.
To check, we can explore the relationship of "nearby" odd integers:
x = 65
y = x+2 = 67
x+y = 132
.
x = 63
y = x+2 = 65
x+y = 128
.
So there is no solution with consecutive odd numbers.
.
If the original problem were to have been x+y = 132, then the solution would have been 65 and 67.
But that was not what was given.