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There appear to be 13 ways. Here they are in {p,n,d,q} form:
{p,n,d,q}
{0,0,0,1}
(0,1,2,0}
{5,0,2,0}
{0,3,1,0}
{5,2,1,0}
{10,1,1,0}
{15,0,1,0}
{0,5,0,0}
{5,4,0,0}
{10,3,0,0}
{15,2,0,0}
{20,1,0,0}
{25,0,0,0}

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F(x) = x^3
F(-9) = (-9)^3 = -9*-9*-9 = -729

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To solve these, use the phrase "letter goes away and number comes to stay as follows:
k(x)=x+20
k(0) = 0 + 20 = 20
k(-17) = -17 + 20 = 3
k (-12) = -12 + 20 = 8
k(14) = 14 + 20 = 34
K (d+17) = (d+17) + 20 = d + 37

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This is also called the exterior triangle theorem. With angle 4 vertical to angle 3, you cannot prove
angle 4 = angle 1 + angle 2. For this to work, angle 4 must be supplementary to angle 3. Here is the proof for that.
STATEMENT 1: triangle ABC
REASON 1: given
STATEMENT 2: angle 1 + angle 2 + angle 3 = 180 degrees
REASON 2: definition of sum of angles of triangle
STATEMENT 3: angle 3 + angle 4 = 180
REASON 3: definition of supplementary angles
STATEMENT 4: angle 1 + angle 2 + angle 3 = angle 3 + angle 4
REASON 4: substitution [2,3]
STATEMENT 5: angle 1 + angle 2 = angle 4
REASON 5: subtraction property of equality

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Primes are not closed under multiplication.
--
Look at each separately:
odd integers: odd x odd = odd [closed]
even integers: even x even = even [closed]
prime numbers: prime x prime = composite [NOT closed]
rational numbers: fraction x fraction = fraction [closed]

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STATEMENT 1: AB bisects CD at M
REASON 1:given
STATEMENT 2: AM congruent BM
REASON 2: definition of bisector
STATEMENT 3: CM congruent DM
REASON 3: definition of bisector
STATEMENT 4: angle CMB congruent angle AMD
REASON 4: definition of vertical angles
STATEMENT 5:triangle CMB congruent triangle AMD
REASON 5: SAS
STATEMENT 6: CB congruent AD
REASON 6: CPCTC
STATEMENT 7: angle CBM congruent angle DAM
REASON 7: CPCTC
STATEMENT 8: CB is parallel to AD
REASON 8: definition of parallel lines.

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The perimeter is
P = 2L + 2W or
80 = 2L + 2W
dividing by 2 we get
(i) 40 = L + W
THe area is
(ii) A = LW
solving (i) for L, we get
(iii) L = 40-W
substituting that in (ii), we get
(iv) A = (40-W)(W)
using completing the square, we get
A = -(W-20)^2 + 400
So, when W = 20, the max area is 400.

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The two equations are:
(i)
(ii)
multiply both equations by 63 to get
(iii)
(iv)
multiply (iii) by 7 and (iv) by -27 to get
(v)
(vi)
adding down, we get

y = 0.
x = 63
{63,0)

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Our equation was

step 1 - rewrite in descending order to get

step 2 - factor to get

step 3 - set each part = 0 and solve to get

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If reduce a price by 35%, you still have 65% left.
So,
.65P = 57.16
dividing we get
p ~ $87.94.
--
checking,
87.94*.35 ~ 30.78
87.94 - 30.78 ~ 57.16.

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We have a fraction of the following type
N/D, where D NOT = 0.
Based on the given information,
D - N = 40
and
N/D = 2/7.
This is a reduced number.
D = 40 + N
by substitution, we get
N / (40+N) = 2/7
cross multiply to get
7N = 80 + 2N
solve for N to get
5N = 80
N = 16
Since N = 16, D = 56
Our fraction is 16/56.

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Let the original number be expressed as

When you reverse the digits, you get

Now the quotient = 2.08 as

We also know that t + u = 7, so t = 7-u. We get

This gives us

cross multiplying, we get

solving for u, we get



original number is 52
reversed number is 25.
quotient is 2.08

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Everything you have is correct. You have created the x,y coordinates. You still have to graph these.

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A solution occurs when x = 4k, y = 2k, and z = 3k where K >=1. So,
x:y:z = 4:2:3

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This is an RTD problem. Here is a table base on given information:
motorist . . . . . . . . .rate . . . . . . . . . .time . . . . . . . . . .distance
. . . . . . . . . . . . . . .. R1 . . . . . . . . . . . .2 . . . . . . . . . .. . 2R1
. . . . . . . . . . . . . . . .R2 . . . . . . . . . . . .3 . . . . . . . . .. . .3R2
. . . . . . . . . . . . . . . R2 . . . . . . . . .. . . . 1 . . . . . . . . .. . .1R2
. . . . . .. . . . . . . . . . .. .. . . . . . . . . . .6 . . . . . .. . . . . . .2444
We see that
(i)
or dividing by 2, we get
(ii)
and that the average rate is
avgR = 2444/6 = 407.33
This means that
(iii)
or
(iv)
substituting (iv) into (ii) we get
(v)
(vii)
solving for R2, we get

This makes R1

--
The rates appear to be the same.

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We set the two sides equal to each other as
5x – 30 = 3x + 10
Solving for x, we get


Now, each side is worth 70.
We have a 70, 70, and x sided triangle.
Using the triangle inequality theorem, we get
70-70 < x < 70+70
or
0 < x < 140.
The greatest integer < 140 is 139.

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To find axis of symmetry, we use -b/2a. So,
-b/2a = -8/2(2) = -8/4 = -2
The axis of symmetry is
X = -2.

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Let x, x+2, x+4 b the 3 consecutive odd integers.
From the given, we get
x+ (x+4) = 78
2x + 4 = 78
2x = 74
x = 37
So, we get
37, 39, 41.

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X is the number of points.
First we have a proportion
5/N = 7/100
Solving, we get n ~ 71.42 or ~ 71.
There are 71 points on this test.
a) write a function, f(x) to represent just the 5 bonus points
ANSWER:

b) Write a function, g(x) to represent just the precent of increase.
ANSWER:

c) Explain the meaning of f(g(x))
ANSWER: The g(x) is a percentage which you then add to a number of points.
d) Explain the meaning of g(f(x))
ANSWER: f(x) is a raw score that you are converting into percentage
e) Does f(g(x)) = g(f(x)) in the problem? Explain
They are not the same.
f(g(x)) = (100x+855)/71
g(f(x)) = (100x + 1000)/71

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Given one die is 5 we could have (5,3) or (3,5). So,
P (sum = 8, given one is 5) = 2/36 = 1/18.

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P(white then red) = 3/8*5/8 = 15/64.
Since we are replacing, the denominator stays the same.

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This is a combination problem. We want 3 of the 4 sevens and don't care about the other cards. SO, we have
p(3 sevens) = (4c3)*(48c2) / (52c5) ~ 0.001736

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So, we know that the average of 10 tests is 84. This is a total of 840 points.
We know the average of 8 tests is 85. his is a total of 680 points.
Let the highest score = H and the lowest score = H - 24.
--
8 tests + 2 dropped tests = 10 tests. SO, we get
680 + 2H - 24 = 840.
solving for H, we get
2H = 840 - 680 +24
2H = 184
H = 92
H-24 = 68
HIs highest grade was 92.

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We know that
a + b = 90
a = b + 10
By substitution we get
b+ 10 + b = 90
2b + 10 = 90
2b = 80
b = 40.

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Lets first find the volume of the cylinder.
We have



--
Now the volume of the cone
We have




--
adding these we get

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Here is our equation
(i) nx^2 – 8x + 10 = 0
We need the discriminant part of the quadratic which is
(ii)
From (i), the D is
(iii)
We want The roots to be equal, so D = 0.
From (iii) we get
(iv)
Solving for n, we get

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We can write an equation based on the given information. Let S = snow white. The 7 dwarfs + snow white = 8 people.

multiply all by 8 to get

subtract S and combine like terms to get

diving, we get
S = $24

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We have the following:
(i)
(ii)
rewrite (i) as
(iii)
this will help with the conversion of J.
--
1H = 1H
1S = 4H
1J = 3S = 12H

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We have the following:

cross multiplying, we get

setting the equation = 0, we get

factoring, we get

solving each we get
a/b = 3/2
a/b = 2/3
So, we have two answers:
a:b = 3:2
a:b = 2:3

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There are 4 jacks, 4 queens, and 4 kings for a total of 12.
P(face card) = 12/52 = 3/13

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I would suggest writing all of these as improper fractions and then multiplying like this:

unfortunately, nothing cancels, so we get

Here it is a matter of division to get

rounded, we get
cu. ft.