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Try:
To make this more natural, remember that the primitive tangent function has period of pi. So we have Tan(-5pi/12) congruent to Tan(7pi/12) mod pi.
Now: What functions do we easily know the value to for trigonometrics? pi/3 is certainly one. So, pi/3=4pi/12---> We need to know 3pi/12=pi/4 ! This is also an easy find.
So, we take Tan(7pi/12)=Tan(3pi/12 + 4pi/12) Now we use your formulae:
Tan(3pi/12 + 4pi/12)=[Tan(pi/4)-Tan(pi/3)]/(1+Tan(pi/4)Tan(pi/3))
Now it remains to remember what Tan(pi/4) and Tan(pi/3) are. We use the definition of Tangent.
Tan(x)=Sin(x)/Cos(x).
So Tan(pi/4)=1.
Tan(pi/3)=(sqrt(3)/2)/(1/2)=sqrt(3)
Finally, substitute into our expression:
(1-sqrt(3))/(1+sqrt(3))
Now, simplify:
==
This could be done quicker, but I hope you can see the full details of what is going on through this procedure.

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Try solving the equations for each term of the sequence.
Eg.
a. 3x-3=23? -> 3x=26 -> x=26/3 Hence x is not a natural number
b. 3+2x=23? -> 2x=20 -> x=10 Hence this is a possible solution
c. 2x-2=23? -> 2x=25 -> x=25/2 Hence x is not a natural number
d. x^2+1=23 -> x^2=22 -> x is irrational and hence not a natural number.

This gives (b) as the only possible solution as we define sequences such that terms are generated by natural numbers. You can easily check the remaining, given terms of the sequence.

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Perhaps it is best to think of "x" as something tangible in real life. Consider x as an orange.
On the left-hand side of the equation you have 1 + 7 oranges. On the right hand side you have 8 oranges. Does 8 = 8? Yes, always. So we will say the solution is x such that x is contained in all real numbers.

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We want to substitute -1 for x, that is x=-1...
f(-1)=5(-1)+4=-5+4=-1

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1) Factor to f(x)=(-2 + x) (-1 - 3 x + x^3)
2) The first evident, real root will be x=2.
3) It should be noted that this will "eventually" resemble a parabola. Moreover, our interest will be around x=2.
4) Techniques of calculus will give us concavity, max, min, inflection.
f'(x)=4x^3-6x^2-6x+5=0 gives criticals of
x -> -1.05274
x -> 0.611731
x -> 1.94101
If you test outside the intervals you will note changes in direction of the slope this is indicative of extrema.
Thus: x=-1.05274 is local minimum
x=0.611731 is local maximum
x=1.94101 is local minimum
Concavity [not entirely necessary]:
f''(x)=12x^2-12x-6=0
gives
x -> -0.366025
x -> 1.36603
Testing intervals again you will find CU on (-inf, -0.366025), CD on (-0.366025, 1.36603), then CU on (1.36603, inf).
This is certainly enough information to produce the following graph:

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30. -4
31. 4i
32. 1/9
33. 3
34. 3i

All of the above can be done by taking what is in the parentheses and forming a square. For instance -sqrt(16)=-sqrt(4^2)=-4.

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I will show you how to apply it:
for:
ax+by=e
cx+dy=f
We will have Cramer's rule giving us:

and

So, just plug in the numbers for the problem we're interested in. Namely, a=1, b=2, c=1, d=-1, e=5, f=3.
The process is done automatically by this web site:

Check:

Solved by pluggable solver: Linear System solver (using determinant)

Solve: Any system of equations: has solution or (x=3.66666666666667, y=0.666666666666667}

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from the first:
y=1-7x
Put that y into the second equation:
3x+2(1-7x)=-9
3x+2-14x=-9
-11x=-11
x=1

If x=1,
y=1-7(1)=-6
check:
7-6=1
3-12=-9
yes.

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The solution before mine is correct, but here is a more suitable graph.

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From the first equation:
z=1-x-2y
Put this into the second equation:
2x+3y+2(1-x-2y)=0
2x+3y+2-2x-4y=0
-y=-2
y=2

If y=2, put that in the third equation:
-x-3(2)+3(1-x-2(2))=1
-x-6+3-3x-12=1
-4x=16
x=-4
Now put y=2 and x=-4 into our z from step 1.
z=1-(-4)-2(2)=1
Giving solution set {-4,2,1}. This also can be solved through techniques of linear algebra (Cramer's Rule, RREF, etc). If you need it solved in a way such as that, send me an E-mail to enabla@gmail.com .

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dot product means multiply each component of one vector with another vector, and add them.
For instance, <1,1,1>*<0,2,3>=2+3=5
So for <1,1>*<-1,1>=0

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<- added 9 to both sides
<- divided both sides by 4
x=+/- 3/2 <-took square root of both sides.

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For v and w to be orthogonal, their dot product must be 0.
(1,1)*(1,b)=1+b=0
b=-1

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First we standardize the score with
z=(x-u)/o
where z is the distance in standard deviations from the mean, x is the score to be standardized, u is the mean of the population, and o is the standard deviation of the population [or sample].
Now, we can suspect from a percentile rank chart that the upper 2% is 2 standard deviations from the mean. In fact, however there is approximately only 100-2.27=97.73 within two standard deviations above the mean. Chances are, however, that your instructor only wants 2 standard deviations above the mean:
z=(x-100)/15
15z+100=x
This gives a score, x, as a function of its distance from the mean[z]. So if we want 2 standard deviations above the mean, 15(2)+100=130. You will need an IQ of 130 to get into MENSA accordingly. Strictly speaking, the score would be a little higher than 130 (in order for to be ahead of 98% of the population) due to the reasoning above.

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(.72+.85+x)/3=.8
1.57+x=2.4
x=.83.
You need an 83%.

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[1] Find the slope of the line connecting the two points. m=(y2-y1)/(x2-x1)=(-3-1)/(7-(-2))=-4/9.
[2] Use point slope form of a line with one point: y-y1=m(x-x1).
y-1=(-4/9)(x-(-2))
y=(-4/9)x+1/9

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y=mx+b gives form with m=slope, b=y-intercept
thus m=-3, b=-10

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[1] Find the slope of the given line:
y=-6x+2
gives m=-6=slope
[2] A perpendicular line will have slope -1/m. In this case, -1/m=1/6
[3] Apply point slope form y-y1=m(x-x1) with point (3,5)=(x1,y1).
y-5=(1/6)(x-3)
y=(1/6)x-1/2+10/2
y=(1/6)x+9/2.
Finished

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Solve for y.
(4x-15)/5=y
(4/5)x-3=y

The slope of a line is m in y=mx+b. b is the y-intercept [b].
Thus, we get m=4/5, y-intercept=b=-3.

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This equation is the same as:
Log((3x-3)/(4(x+1))=0
Which is moreover the same as:
4=(3x-3)/(x+1)
4x+4=3x-3
x=-7

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add 8 to both sides of the equation.
b=8.

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Yes, 3a-2b is the correct simplification.

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You can actually just subtract the first equation from the second in this case.
This gives:
0=x-2
2=x

And if x=2, y=4 by back substitution.


You could also solve by substitution, graphing, or other techniques from linear algebra (such as Cramer's Rule after some slight rearrangement).

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3 consecutive odd integers:
a, a+2, a+4
a+a+2=5+a+4
a=7
So our largest is 7+4=11.

Check:
7+9=16 which is 5 more than 11.

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I will get you started on this problem:
a^3+b^3=(a+b)(a^2-ab+b^2)
and your problem here is x^3+2^3. Apply the rule.
Rectangular form is simply giving the answers in (x,y).

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3 consecutive odd numbers:
a
a+2
a+4
3a+6>375
3a>369
a>123

Now, if we want strictly more than 375, we will take a=125 to be our first odd number. This gives 125,127,129 as the numbers.

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a+a^2=30
a^2+a-30=0
(a-5)(a+6)=0
so a=5 or a=-6
a must be an integer therefore a=-6 is excluded.
a=5 is the only possible number.


Check:
5+25=30.
yes

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3 numbers:
a
a+1
a+2
words to symbols:
a(a+1)-3(a+2)=57
a^2+a-3a-6=57
a^2-2a-63=0
(a+7)(a-9)=0
so a=-7 or a=9. We want a to exist in the set of integers. Thus, we exclude a=-7 as a solution.
The numbers, then, are 9, 10, 11.
Check:
9(10)-3(11)=90-33=57
yes

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3 consecutive integers:
a
a+1
a+2
words to symbols:
a+a+1=a+2+6
a=7

So the numbers are 7, 8, 9.
check:
first two numbers added: 15
6 more than 9: 15
yes

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What is the third root of 1000? 10. What is 10^5? 100,000.