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function is defined as:

f(x) = cos^2(x) + 1/csc(2x)

what is f(pi/6)

you replace x with pi/6 and solve.

cos^2(pi/6) = [cos(pi/y)]^2= (.866025)^2 = .75
1/csc(2x) = sin(2x) = sin(2pi/6) = .8660254038

cos^2(pi/6) + 1/csc(2x) = .75 + .86... = 1.616025404

if you convert this to degrees it might be easier to see.

you convert pi/6 to degrees by multiplying it by 180 and dividing it by pi.

pi/6 * 180/pi = 180/6 = 30 degrees.

cosine 30 degrees = sqrt(3)/2

sine 60 degrees = sqrt(3)/2

cos^2(30) = (sqrt(3)/2)^2 = 3/4

1/csc(2x) = sin(2x) because sine = 1 / cosecant and cosecant = 1 / sine.

x = 30 so 2x = 60 degrees.

sine(60) = sqrt(3)/2

cos^2(30) + sin(60) = 3/4 + sqrt(3)/2 = (3 + 2*sqrt(3))/4 = 1.616025404.

this is the same answer we got before using radians.

the graph of the equation of cos^2(x) + 1/csc(2x) is shown below.



getting it to graph properly is a challenge, but if you know a few tricks, it can be done.

first trick:

cos^2(x) is the same as (cos(x))^2
first you find cos(x) and then you square it.

second trick:

csc(2x) = 1/sin(2x) and sin(2x) = 1/csc(2x) by definition.
sin(2x) is the sine of 2 times the angle.
if the angle is 30 degrees, then 2x is 60 degrees.

last, but not least, is that the graphing software deals in radians and not degrees.

on the graphing software, the value of x = pi/6 radians will show up as .523598776 because pi/6 is equal to 3.141592654... / 6 which is equal to .5235.....

if you're using a scientific calculator, make sure to set it to radians before trying to solve this problem directly.

if you want to work with degrees, make sure to convert the radians to degrees before working the problem.

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should be x = 7
factor the quadratic equation as follows:
(x-a)*(x+b) = 0
you want an a and a b such that b-a = 2
look at the factors of a*b that equal 63.
you get:
63*1
21*3
7*9
looks like 7 and 9 will do it because 9-7 = 2
set b = 9 and a = 7 and you get:
(x-7) * (x+9) = 0
results in x = 7 or x = -9
positive x-intercept is 7.
graph of the equation of x^2 + 2x - 63 confirms it as shown below:

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you have a right triangle.
one angle other than the right angle.
one side opposite the angle.
you can use the sine function to get the hypotenuse or you can use the tan function to get the side opposite the other angle.
sine (37) = 16 / hypotenuse
multiply both sides of this equation by the hypotenuse and divide both sides of this equation by sine (37) to get:
hypotenuse = 16 / sine(37)
use your calculator to find the sine(37) and then solve for the hypotenuse.
similarly.
tangent (37) = 16 / side opposite the other angle.
multiply both sides of the equation side opposite the other angle and divide both sides of the equation by tangent (37) to get:
side opposite the other angle = 16 / tangent (37).
if you do this correctly, you will get:
hypotenuse = 26.58624226
side opposite the other angle = 21.23271715
to confirm you did this correctly, use the pythagorean formula to get:
(16)^2 + (21.23271715)^2 = (26.58624226)^2
this results in:
706.0282774 = 706.8282774 confirming the values for the hypotenuse and the other leg are good.

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a graph of the equation of y = 2x-3 is shown below:

your table will look like this:

x             y = 2x-3
-3              -9
-2              -7
-1              -5
 0              -3
 1              -1
 2               1
 3               3

all you're doing is replace x in the equation of y = 2x-3 with the value of x.
when x = -3, y = 2x-3 becomes y = 2(-3) - 3 which becomes y = -6 - 3 which becomes y = -9
similarly, when x = 3, y = 2x-3 becomes y = 2(3) - 3 which becomes y = 6 - 3 which becomes y = 3.
etc.

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the present value of 1.5 million is equal to 1.5 million if it is all payed up front.
the present value of 1.425 million is equal to 1.425 million if it is all payed up front.
the present value of .5 million payed up front and .5 million payed in 6 months and .5 million payed in 12 months would be equal to:
.5 million plus .5 million / (1.01)^6 plus .5 million / (10.1)^12 which would be equal to:
.5 million plus .4710226176 million plus .4437246126 which is equal to:
1.41474723 million.
the best you're going to do is if the 1.5 million is payed up front.
the next best you're going to do is if you accept the 1.425 million offer.
the last best you're going to do is if you accept the 3 payment plan and don't charge interest on the deferred payments.
if you accept the deferred payment offer but charge 12% interest compounded monthly on the deferred payments, then you will make the following:
.5 million on the part that is paid up front.
.5 million * (1.01)^6 on the part that is paid in 6 months.
.5 million * (1.01)^12 on the part that is paid in 12 months.
the present value of that is equal to:
.5 million on the first part.
plus .5 million * (1.01)^6 / (1.01)^6 = .5 million on the second part.
plus .5 million * (1.01)^6 / (1.01)^12 = .5 million on the third part>
this will total 1.5 million which is the original asking price.
the payment of 1.5 million up front is equivalent to paying in 3 parts with interest being charged for the 3 parts, so you will make the same money.
the bottom line is:
1.5 million up front is a present value of 1.5 million.
1.5 million in 3 payments and being charged 1% per month for the deferred payments still gives you a present value of 1.5 million.
1.425 million present value is next in line.
1.41 million present value is next in line.
the deferred payment plan that charges 1% interest for each deferred month of payment is equivalent to 1.5 million up front and is the preferred plan if deferred payments are to be considered.

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you don't have enough information to come up with a single answer.
you can solve this problem, but you will need to know either:
total time
or:
total distance
the basic formula is RT = D
R = rate of travel
T = time
D = distance
i don't know what you mean by kms.
if that's kilometers per second then it's not realistic.
the average person walks 3 miles per hour which is roughly equivalent to 5 kilometers per hour.
that would be 5kph.
i will assume 5kph.
assuming that, this is how it will work out.
the distance traveled is shown as follows:
day 1 equals D
day 2 equals D/2
day 3 equals (D/2)/3 which is equal to D/6
day 4 equals (D/6)/4 which is equal to D/24
the total distance traveled over the 4 days is:
D + D/2 + D/6 + D/24
this comes out to be 41D/24 total.
D = 24D/24
D/2 = 12D/24
D/6 = 4D/24
D/24 = D/24
since they now have a common denominator, just add up the numerators and divide the total by the denominator to get (24 + 12 + 4 + 1)/24 which is equal to:
41D/24
the rate is always the same (5kmph)
the total time is calculated by use of the formula to get:
total time * rate = total distance
we solve for time to get:
total time = total distance / rate
since total distance is equal to 41D/24, and since rate is equal to 5kmph, this becomes:
total time = (41D/24) / 5 which becomes 41D/(24*5) which becomes .3416666667D
your final equation is:"
total time = .3416666667D
you have 2 unknowns in one equation.
best you can do is make an assumption about total time or about total distance.
when you know one of them, you can find the other.
you can't, however find both.
not with the information given.
i assumed the total time was 4 hours.
the equation of:
total time = .3416666667D becomes:
4 = .3416666667D
i then solved for D to get:
D = 4/.3416666667 = 11.70731707 kilometers
that distance only applies if the total time is 4 hours.
change the total time and the distance changes.
assuming that the total time is 4 hours, the day by day breakdown was:

day              distance (km)          rate (kmph)          time (hours)
1                  11.70731707          5                     2.341463415
2                  5.853658537          5                     1.170731707
3                  1.951219512          5                     0.3902439024
4                  .487804878           5                     0.0975609756
total              20                   ---                   4

total distance was 20 kilometers and total time was 4 hours.
distance traveled the first day was 11.70731707 kilometers
this value is only accurate if the total time was assumed to be 4 hours.
once again, there is not enough information provided to give an exact answer.
you have too many unknowns and not enough equations.
to get 1 answer, you need to have as many equations as you have unknowns (variables).
you had 2 unknowns (time and distance) with only 1 equation (rate * time = distance).

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(3x^3y^-2)(-2xy^3)^-1 is your expression.
since, in general, x^-a = 1/x^a, this expression can become:
(3x^3y^-2) / (-2xy^3)
since, in general, (a*b)/(c*d) is the same as (a/c) * (b/d), this expression can become:
-(3/2) * (x^3/x) * (y^-2/y^3)
since, in general, x^a / x^b is equal to x^(a-b), this expression can become:
-(3/2) * (x^3-1) * (y(-2-3) which can be simplified to:
-(3/2) * x^2 * y^-5
since, in general, x^-a = 1/x^a, this expression can become:
-(3/2) * x^2 * 1/y^5
since, in general, (a/c) * (b/d) is the same as (a*b) / (c*d), this expression can become:
(-3*x^2*1) / (2*y^5) which can be simplified to:
(-3x^2)/(2y^5)
that's your solution.
you can confirm the solution is good by picking value for x and for y and using those values in the original equation and the final equation.
if the solution comes out the same, then you did good.
you can use your calculator to perform the calculations.
i did with the values of x = 5 and y = 7.
i got:
original equation equals: .0022312132
final equation equals: .0022312132
they're the same so the original equation is equivalent to the final equation so we're good.

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log(4,x+2) - log(4,x-1) = 1
since, in general, log(x) - log(y) = log(x/y), this equation becomes:
log(4,(x+2)/(x-1)) = 1
since, in general, log(b,x) = y if and only if b^y = x, this equation becomes:
4^1 = (x+2)/(x-1) which is the same as:
(x+2)/(x-1) = 4
multiply both sides of this equation by (x-1) to get:
(x+2) = 4*(x-1)
simplify to get:
x+2 = 4x-4
subtract x from both sides of this equation and add 4 to both sides of this equation to get:
6 = 3x which is the same as:
3x = 6
divide both sides of this equation by 3 to get:
x = 2
replace x with 2 in your original equation.
log(4,x+2) - log(4,x-1) = 1 becomes:
log(4,(4) - log(4,1) = 1
this is true if and only if 4^1 = 4
since 4^1 is equal to 4, this confirms the solution of x = 2 is good.
you can also confirm it by solving it using the LOG function of your calculator.
to do that, you need to convert the equation as follows:
log(4,(4) - log(4,1) = 1 becomes:
LOG(4)/LOG(4) - LOG(1)/LOG(4) = 1
use the log function of your calculator to solve to get:
you get:
1 - 0 = 1 which becomes 1 = 1 which is true, confirming x = 2 is good.
LOG(4) / LOG(4) is equivalent to log(10,4) / log(10,4)
LOG(1) / LOG(1) is equivalent to log(10,1) / log(10,4)
since, in general, x/x = 1, LOG(4)/LOG(4) = 1
since log(10,1) = y if and only if 10^y = 1, this occurs only when y = 0 which makes the expression log(10,1) equivalent to 0 which is confirmed when you use your calculator to solve for LOG(1). you get 0 as the answer.

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problem number 1:

(x^2 - 6x)^2 - 11(x^2-6x)-80 = 0

let y = x^2 - 6x
equation becomes:
y^2 - 11y - 80 = 0
this is a quadratic equation that can be solved in the normal manner that you solve quadratic equations.

80 = 1 * 80 or 2 * 40 or 4 * 20 or 5 * 16 or 10 * 20
1 +/- 80 = 81 or 79
2 +/- 40 = 42 or 38
4 +/- 20 = 24 or 16
5 +/- 16 = 21 or 11 *****
10 +/- 20 = 30 or 10
looks like the combination of 5 * 16 is going to work because 5 * 16 = 80 and 16 - 5 = 11
the 80 is the c term and the 11 is the b term.
our factors are going to be (y + 5) * (y - 16) = 0
if you multiply these factors out you will get:
y^2 + 5y - 16y - 80 which will be equal to y^2 - 11y - 80 which is the quadratic equation we are trying to factor after we let (x^2-6x) equal to y.
since (y+5) * (y-16) = 0, we set each equal to 0 to solve for y.
we get:
y + 5 = 0 which becomes y = -5
y - 16 = 0 which becomes y = 16
those are our solutions for y.
we now need to convert those solutions of y to solutions of x.
we had originally set y = (x^2-6x), so we now substitute (x^2 - 6x) for y to get:
x^2 - 6x = -5
and:
x^2 - 6x = 16
we now need to solve for x.
x^2 - 6x = -5 becomes x^2 - 6x + 5 = 0 after we add 5 to both sides of the equation.
x^2 - 6x = 16 becomes x^2 - 6x - 16 = 0 after we subtract 11 from both sides of the equation.
x^2 - 6x + 5 factors are (x-5) * (x-1)
x^2 - 6x - 16 factors are (x-8) * (x+2)
we set each of these factors = to 0 to get:
x = 5
x = 1
x = 8
x = -2
those are our possible solutions.
we need to confirm those solutions are good by substituting for x in our original equation.
that equation is:
(x^2 - 6x)^2 - 11(x^2-6x)-80 = 0
i used a graphing calculator to confirm those solutions were good.
they are.
without a graphing calculator you need to substitute each of those values for x in turn in the equation and solve the equation to see if the result is 0.
example:
when x = 1, original equation of (x^2 - 6x)^2 - 11(x^2-6x)-80 = 0 becomes:
(1^2 - 6(1))^2 - 11(1^2 - 6(1)) - 80 = 0
this simplifies to:
(1 - 6)^2 - 11(1 - 6) - 80 = 0 which further simplifies to:
(-5)^2 - 11(-5) - 80 = 0 which further simplifies to:
25 + 55 - 80 = 0 which further simplifies to:
80 - 80 = 0 which is true, confirming the value of 1 for x is good.
your solutions to this equation are:
x = -2
x = 1
x = 5
x = 8

problem number 2:

(x-1)/(x-3)>0

you can try to solve this by doing the following:
(x-1) / (x-3) > 0
multiply both sides of the equation by (x-3) to get:
(x-1) = 0
solve for x to get x > 0
that, however, leads to only a partial solution if at all.

these types of equations need to address the numerator and the denominator separately.

find out when the numerator is equal to 0 and find out when the denominator is equal to 0.
the numerator is equal to 0 when x = 1
the denominator is equal to 0 when x = 3

the fraction (x-1) / (x-3) will be positive when the numerator is plus and the denominator is plus or when the numerator is minus and the denominator is minus.

the numerator is positive when x > 1 and the denominator is positive when x > 3.
since 3 > 1, then both numerator and denominator will be positive when x > 3.

the numerator is negative when x < 1 and the denominator is negative when x < 3.
since 1 < 3, then both numerator and denominator will be negative when x < 1

put these facts together and the fraction will be positive when x < 1 and when x > 3.

you can graph this equation to visually see what's going on.
graph the equations of y = (x-1) / (x-3) as shown below:

as you can see from the graph, there is a vertical asymptote at x = 3.
this happens because the denominator of the equation becomes 0 when x = 3.
i drew a vertical line there to show you where that happens.
you can also see the the lines of the equation are positive when x is smaller than 1 and when x is greater than 3.

problem number 3:

absolute value of (x^2+x-15) = 15

the definition of an absolute value says that:

if absolute value of (x) = y, then:
x = y if x is positive and:
-x = y if x is negative.

x represents an expression that is within the absolute value signs.
those are vertical lines as shown below:
absolute value of (x) is shown as |x|.

your equation is:
absolute value of (x^2+x-15) = 15
this can also be shown as:
|x^2 + x - 15| = 15

the expression within the absolute value signs is x^2 + x - 15

when that expression is positive, you get:
x^2 + x - 15 = 15
solve as you would any normal quadratic equation.
subtract 15 from both sides of the equation to get:
x^2 + x - 30 = 0
factor to get:
(x-5) * (x+6) = 0
solve for x to get:
x = 5 or x = -6

when that expression is negative, you get:
-(x^2 + x - 15) = 15
multiply both sides of this equation by -1 to get:
x^2 + x - 15 = -15
add 15 to both sides of this equation to get:
x^2 + x = 0
factor to get:
x * (x+1) = 0
solve for x to get:
x = 0 or x = -1

put both solutions together to get:
x = 0 or x = -1 or x = 5 or x = -6
these are the possible solutions to your equation.

you need to confirm these solutions are good by substituting in the original equation.
that equation is:
|x^2 + x - 15| = 15

when x = 0, |x^2 + x - 15| = 15 becomes:
|0 + 0 - 15| = 15 which becomes:
|-15| = 15 which is true by definition of absolute value.

when x = -1, |x^2 + x - 15| = 15 becomes:
|(-1)^2 + (-1) - 15) = 15 which becomes:
|1 - 1 - 15| = 15 which becomes:
|-15| = 15 which is true by definition of absolute value.

when x = 5, |x^2 + x - 15| = 15 becomes:
|5^2 + 5 - 15| = 15 which becomes:
|25 + 5 - 15| = 15 which becomes:
|30 - 15| = 15 which becomes:
|15| = 15 which is true by definition of absolute value.

when x = -6, |x^2 + x - 15| = 15 becomes:
|(-6)^2 + (-6) - 15| = 15 which becomes:
|36 - 6 - 15| = 15 which becomes:
|30 - 15| = 15 which becomes:
|15| = 15 which is true by definition of absolute value.

definition of absolute value states:
|x| = y means:
x = y if x is positive.
-x = y if x is negative.

in our equation of |-15| = 15, 15 is negative, so we get -(-15) = 15 which becomes 15 = 15.
in our equation of |15| = 15, 15 is positive, so we get 15 = 15 which is self evidently true.

you can also graph the equation of |x^2 + x - 15| = 15 so see what it looks like on a graph.
subtract 15 from both sides of that equation to get:
|x^2 + x - 15| - 15 = 0
replace 0 with y in that equation to get:
y = |x^2 + x - 15| = 0 and graph that equation.
you will get:

we'll take a closer in look so you can see the points where the graph crosses the x-axis.
those points are your solution points.

you can see that the graph crosses the x-axis at x = -6, -1, 0, and 5.
this graph is the graph of the absolute value of (x^2 + x - 15) - 15.
this is not the same as the graph of x^2 + x - 15 - 15 = 0
the graph of that equation looks like this:

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T = total number of bottles in the cooler.
x = number of bottles of grapefruit juice.
x = number of bottles of apple juice.
2x + 4 = number of bottles of orange juice.
T = x + x + 2x + 4
combine like terms to get:
T = 4x + 4
number of bottle of orange juice is equal to 4x + 4 - 2x
the number of bottles of orange juice is equal to 2x + 4.
you don't have enough information to narrow it down better than that.
you either need to know the total number of bottles or the number of bottles of apple or graph juice.

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interest rate is 8% per year = .08 per year
total repayment is 4640 on a loan of 4000.
total interest to be paid is 640.
total interest each year is calculated at .08 * 4000 = 320 per year.
640 / 320 = 2 years when the total amount on the loan equals 4640 which is when the loan matures.
paying the loan before 2 years is up requires less interest to be paid.
paying the loan after 2 years is up requires more interest to be paid.
total due is 4640 in exactly 2 yeara.

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A university bookstore recently sold a wire bound graph-paper notebook for $2.50, and a college-ruled notebook for $2.30. At the start of spring semester, a combination of 50 of these notebooks were sold for a total of $118.60. How many of each type were sold?

w = number of wire bound notebooks.
c = number of college ruled notebooks.
total of 50 notebooks were sold for $118.60.
your equations are:
2.50w + 2.30c = 118.60
w + c = 50
these are 2 equations that need to be solved simultaneously.
you can use substitution.
from the second equation, solve for w to get:
w = 50 - c
substitute for w in the first equation to get:
2.50*(50-c) + 2.30c = 118.60
simplify to get:
125 - 2.50c + 2.30c = 118.60
combine like terms to get:
125 - .20c = 118.60
subtract 125 from both sides of the equation to get:
.20c = -6.4
divide both sides of the equation by -.20 to get:
c = -6.4/-.20 = 32
since w + c = 50, this means that w equals 18.
you have w = 18 and c = 32
w + c = 18 + 32 = 50
2.50*w + 2.30*c = 118.60 becomes:
2.50*18 + 2.30*32 = 118.60
answer is good.
18 wire bound notebooks are sold
32 college ruled notebooks are sold.

A bricklayer and an electrician together spend 90 hours working on a new house. If the bricklayer earns $12 per hour, the electrician earns $16 an hour, and the owner pays them a total of $1350 for their work, how many hours does each worker spend on the house?

x = number of hours the bricklayer works
y = number of hours the electrician works
x + y = 90
12x + 16y = 1350
you have 2 equations that need to be solved simultaneously.
you can solve by substitution.
solve for y in the first equation to get:
y = 90 - x
substitute for y in the second equation to get:
12x + 16*(90-x) = 1350
simplify to get:
12X + 1440 - 16X = 1350
combine like terms to get:
-4x + 1440 = 1350
subtract 1440 from both sides of the equation to get:
-4x = 1350 - 1440 = -90
divide both sides of the equation by -4 to get:
x = 22.5
since x + y = 90, this means that y is equal to 90 - 22.5 = 67.5
x = 22.5
y = 67.5
x + y = 90
12*x + 16*y = 1350
12*22.5 + 16*67.5 = 270 + 1080 = 1350
answers are good.
bricklayer works 22.5 hours
electrician works 67.5 hours

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the area of the base remains the same.
that appears to be a given.
only the height has changed.
the volume of a prism is equal to the height * the area of the base.
assume the area of the base is y and assume the height is x, then the original volume is equal to x*y.
now the height is increased by 3 units.
that makes the height equal to (x+3).
the new volume is now equal to (x+3)*y which becomes xy + 3y
the new volume is xy + 3y
the old volume is xy
it appears that the new volume is more than the original volume by 3 times the area of the base.
i believe that's your answer and it would be worded that way.
3 times the area of the base is the additional volume measured in cubic units.
the area of the base is measured in square units.

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your first 2 equations are in slope intercept form already.
they are:
y = 2x + 3 and y = -x - 4
since they are both equal to y, you can set them equal to each other to get:
2x + 3 = -x - 4
add x to both sides of the equation and subtract 3 from both sides of the equation to get:
3x = -7
divide both sides of the equation by 3 to get:
x = -7/3
substitute -7/3 for x in either equation to solve for y.
use y = 2x + 3 which becomes:
y = -14/3 + 3 which becomes:
y = -14/3 + 9/3 which becomes:
y = -5/3
substitute for x and y in the other equation of y = -x - 4 to get:
-5/3 = -(-7/3) - 4 which becomes:
-5/3 = 7/3 - 12/3 which becomes:
-5/3 = -5/3 confirming the values for x and y are good.

your second set of equations is in standard form which can be solved using substitution or elimination.

the equations are:
2x + 3y = 8 and 3x + 2y = 7

you could solve for y in both equations and use the method described above but that is not necessary.

solving by substitution:
your equations are:
2x + 3y = 8
3x + 2y = 7
solve for x or y in one of the equations and then use that value to substitute in the other equation as shown below:
take the first equation and solve for y:
2x + 3y = 8
subtract 2x from both sides of the equation to get:
3y = 8 - 2x
divide both sides of the equation by 3 to get:
y = (8-2x)/3
substitute that value for y in the second equation.
3x + 2y = 7 becomes:
3x + 2*(8-2x)/3 = 7
simplify to get:
3x + 16/3 - 4x/3 = 7
multiply both sides of the equation by 3 to get:
9x + 16 - 4x = 21
combine like terms to get:
5x + 16 = 21
subtract 16 from both sides of the equation to get:
5x = 5
divide both sides of the equation by 5 to get:
x = 1
use that value for x in the first equation to get:
2x + 3y = 8 becomes:
2 + 3y = 8
subtract 2 from both sides of the equation to get:
3y = 8-2 which becomes:
3y = 6
divide both sides of the equation by 3 to get:
y = 2
substitute 1 for x and 2 for y in the second equation to get:
3x + 2y = 7 becomes:
2 + 4 = 7 which becomes:
7 = 7
this confirms the values of 1 for x and 2 for y are good.
these values apply to both equations at the same time (simultaneously).
they are a common solution to both equations.
2x + 3y = 8 becomes 2 + 6 = 8 which becomes 8 = 8.
3x + 2y = 7 becomes 3 + 4 = 7 which becomes 7 = 7.
x = 1 and y = 2 is a common solution to both equations.

solving by elimination:
your equations are:
2x + 3y = 8
3x + 2y = 7
you are going to add these equations together or you are going to subtract one of these equations from the other to eliminate one of the variables which will allow you to solve for the remaining variable.
the process that allows you to do this is the ability to multiply or divide both sides of each equation without changing the equality that is already there as represented by the equal sign.
in this system of equations, i want to eliminate the x variable.
in order to do that i need to multiply the first equation by 3 and the second equation by 2.
after doing, that my equations become:
6x + 9y = 24 (both sides of first equation multiplied by 3)
6x + 4y = 14 (both sides of second equation multiplied by 2)
now i will subtract the second equation from the first to get:
5y = 10
i will now divide both sides of this equation by 5 to get:
y = 2
i will now use the value of 2 for y in either equation to solve for x.
taking the first original equation (use the original equation for this just in case you messed up when you multiplied the original equation to get the modified equation).
the original equations are:
2x + 3y = 8 (first equation)
3x + 2y = 7 (second equation)
substituting 2 for y in the first equation gets:
2x + 3y = 8 becomes:
2x + 6 = 8
subtract 6 from both sides of the equation to get:
2x = 2
divide both sides of the equation by 2 to get:
x = 1
substitute 2 for y and 1 for x in the second equation to get:
3x + 2y = 7 becomes:
3 + 4 = 7 which becomes:
7 = 7
this confirms the values for x and y are good.
your solution is x = 1 and y = 2
this solutions applies to both equations simultaneously.

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current ages are 12, 40, and 38
you want to know in how many years the sum of these ages will be 150.
let y equal the number of years from now.
your equation becomes:
12 + y + 40 + y + 38 + y = 150
this becomes:
90 + 3y = 150
subtract 90 from both sides of this equation to get:
3y = 60
divide both sides of this equation by 3 to get:
y = 20
in 20 years, the sum of your ages will be 150.
12 + 20 = 32
40 + 20 = 60
38 + 20 = 58
32 + 60 + 58 = 150
your equation didn't look quite right but you got the right answer so you must have been thinking correctly even though someone who looked at your equation might not have thought so in the beginning.
your problem in your equation is you were using y to represent 3 different ages at one time.
normally that would be shown as 3 different variables.
assume the following:
let y1 = the age of the first person
let y2 = the age of the second person
let y3 = the age of the third person
let a = the number of years
then your equation would have become:
(y1 + a) + (y2 + a) + (y3 + a) = 150
this would then have been simplified to:
y1 + y2 + y3 + 3a = 150
you could then substitute 12 for y1 and 40 for y2 and 38 for y3 to get:
12 + 40 + 38 + 3a = 150 which would have led to:
90 + 3a = 150
you would then have solved as before using standard equation solving procedures.

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your equation is:
f(x) = 4cos(3x)+1
you want to shift it to the right by pi/12.
this means that if x is equal to pi/12, you want to get the cosine of x = 0.
the result of that is that the cosine of 0 is now the cosine of pi/12 which means that value has been shifted to the right by pi/12.
the general formula for a trigonometric function is:
y = a*cos(b(x-c))+d
a is the amplitude.
b is the frequency
c is the horizontal shift
d is the vertical shift of the center line.
your equation has an amplitude of 4 and a vertical shift of 0.
your equation has a frequency of 3 which means the period is equal to 2*pi/3 which makes it equal to (2/3)*pi instead of the 2*pi that it normally is.
based on what you have, the standard form of your equation would be:
f(x) = 4*cos(3*(x-0))+1
a graph of your equation looks like this:

a vertical line is shown at x = pi/12 to show you where your shift will occur.
since you want to shift the graph to the right, then the value of cosine at x = 0 will then become the value of cosine at x = pi/12.
to do this, your equation has to go from:
f(x) = 4*cos(3*(x-0))+1
to:
f(x) = 4*cos(3*(x-(pi/12)))+1
the graph of that equation looks like this:

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a circle has no vertex.
if you mean an angle whose vertex lies on the circumference of a circle, then you are talking about an inscribed angle.
a central angle has its vertex at the center of the circle.
the degree measure of a central angle is equal to the degree measure of its intercepted arc.
the degree measure of an inscribed angle is equal to 1/2 the degree measure of its intercepted arc.
the intercepted arc is the part of the circumference of the circle that is between the sides of the angle formed.
try this reference to learn more.
http://www.mathopenref.com/circleinscribed.html

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the equation is y = 9
the value of y will always be 9 because the line is horizontal.
the value of x can be anything, including x = -5.
the point (-5,9) is on the line.

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4x-3y=2
5x+3y=6
12x-9y=6
10x+6y=2

y = (4/3)x - 2/3
y = (-5/3)x + 2
y = (4/3)x - 2/3
y = -5/3)x + 1/3

2 of the lines have the same slope and the same y intercept so they are identical. if you graph these equations, they will look like one line rather than 2.
2 of the lines have the same slope and different y intercept so they are parallel.
you have 1 set of identical lines and 1 set of parallel lines that will intersect in 2 separate points on a graph.
for there to be a common solution, all 4 lines would have had to intersect in one common point on the graph.
since there is not one common solution for all 4 equations, this system of equations has no solution.

a graph of the lines will show the 2 points of intersection.

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a = 20 = number of horror novels.
b = 15 = number of regular novels.
c = 25 = number of mystery novels.
total number of books is equal to 20 + 15 + 25 = 60.
you pick 4 books at random.
you need to pick at least 1 of each type.
3 things have to happen.
you do not pick 0 type a or you do not pick 0 type b or you do not pick 0 type c.

the probability of picking 0 type a is equal to 40/60 * 39/59 * 38/58 * 37/57 = P(40,4)/P(60,4).
the probability of picking 0 type b is equal to 45/60 * 44/59 * 43/58 * 42/57 = P(45,4)/P(60,4).
the probability of picking 0 type c is equal to 35/60 * 34/59 * 33/58 * 32/47 = P(35,4)/P(60,4).

the combined probability of picking 0 type a or picking 0 type b or picking 0 type c is equal to:
(P(40,4) + P(45,4) + P(35,4)) / (P(60,4)
this comes out to be equal to .600336317
this is the probability that you will get {0 type a or 0 type b or 0 type c}.

the probability that you will not get {0 type a or 0 type b or 0 type c} is equal to 1 minus .600336317 which is equal to .399663683.

not getting 0 of each type means that you will get at least 1 of each type.

this is difficult to confirm that the formulas work because the possible combinations are 81 and each one of those possible combinations has its own probability, the sum of which must be equal to 1.

it's easier to show if you have 2 choices of books and are drawing 3 books.
in this case, there are 8 possible combinations so it's easier to show the probability of each combination and to show that the sum of the probabilities is equal to 1.

it's also easier to then confirm that the formula of 1 minus the probability of {0 type a or 0 type b} provides the correct answer.

those details are shown below:

number of type a	40		
number of type b	20		
total	60		
			
summary                         probability	formula used	
p(0 type a) =                   0.033313852	equals 20/60*19/59*18/58	***
p(1 type a and 2 type b) =	0.222092344	equals 3*40/60*20/59*19/58	
p(2 type a and 1 type b) =	0.455873758	equals 3*40/60*39/59*20/58	
p(0 type b) =                   0.288720047	equals 40/60*39/59*38/58	***
total probability	        1		
			
details	                        probability	formula used	
aaa	                        0.288720047	equals 40/60*39/59*38/58	***
aab	                        0.151957919	equals 40/60*39/59*20/58	
aba	                        0.151957919	equals 40/60*20/59*39/58	
abb	                        0.074030781	equals 40/60*20/59*19/58	
baa	                        0.151957919	equals 20/60*40/59*39/58	
bab	                        0.074030781	equals 20/60*40/59*19/58	
bba	                        0.074030781	equals 20/60*19/59*40/58	
bbb	                        0.033313852	equals 20/60*19/59*18/58	***
total probability	        1		

the summary shows you that the probability of getting 0 type a is the same as the probability of getting 3 type b.
it also shows you that the probability of getting 0 type b is the same as the probability of getting 3 type a.
the probability of getting 3 type a or 3 type b is shown in the details.

the probability of getting at least 1 of each type is equal to 1 minus the probability of getting 0 type a or 0 type b.

the same procedure is used in the larger numbers where we picked 4 books from a selection of 3.
finding the number of 0 type a combination was more complicated in this case because there were 16 combinations where 0 type a were picked and 16 combinations where 0 type b were picked and 16 combinations where 0 type c were picked.
for example, the possible combinations where type a books were not picked are:
bbbb
bbbc
bbcb
bbcc
bcbb
bcbc
bccb
bccc
cbbb
cbbc
cbcb
cbcc
ccbb
ccbc
cccb
cccc
this made it difficult to show the details because the individual calculations became onerous to say the least.
it was much easier to reduce the number of possibilities in order to show you how it worked.

all of these combinations, however, involved 1 minus the probability of getting 0 type a or 0 type b or 0 type c.
the same calculations in the simpler example produced the same result, i.e. 1 minus probability of 0 type a minus probability of 0 type b gave the result of at the probability of least 1 of each type.

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3x^2 + 3x^2 = (3+3) * x^2 = 6x^2

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1/3(6)^(-4x)+1=6
i think your equation is this:


if so, then you would solve it as follows:

subtract 1 from both sides of the equation and then multiply both sides of the equation by 3 to get:



take the log of both sides of this equation to get:



since log(x^a) = a*log(x), this equation becomes:



divide both sides of this equation by to get:



this becomes:

-4x = 1.511391594

divide both sides of this by -4 to get:

x = -.0377847899

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you can't.
you need more information.
1 side is not enough.
2 sides of the triangle and an included angle would be enough.
3 sides of the triangle would be enough.
other combinations might work but 1 side by itself is not enough.

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if it has a common ratio then it's geometric.
if it has a common difference then it's arithmetic.
your equation is:
Tn = 24 x (-1/2)^(n-1)
your common ratio looks like it's (-1/2)
let's see how this works.
T1 = 24 * (-1/2)^0 = 24 * 1 = 24
T2 = 24 * (-1/2)^1 = 24 * -1/2 = -12
T3 = 24 * (-1/2)^2 = 24 * 1/4 = 6
T4 = 24 * (-1/2)^3 = 24 * -1/8 = -3
if you look at your number sequence you will see that the common ratio is (-1/2)
24 * (-1/2) = -12 * (-1/2) = 6 * (-1/2) = -3, etc.

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if the probability of failure of one component is .06, then the probability of success of one component is 1 - .06 = .97
if you take a sample of n components, then the formula for the number of failures in that sample would be:
p(x failures) = C(n,x) * p^x * q^(n-x)
C(n,x) equals the combination formula of n! / (x! * (n-x)!)
p^x equals the probability of failure raised to the power of x.
q^(n-x) equals the probability of success raised to the power of (n-x).
example:
you draw a sample of 3 components.

p(0 failures) = C(3,0) * .06^0 * .94^3 which equals .830584
p(1 failure)  = C(3,1) * .06^1 * .94^2 which equals .159048
p(2 failures) = C(3,2) * .06^2 * .94^1 which equals .010152
p(3 failures) = C(3,3) * .06^3 * .94^0 which equals .000216
total of all possible probabilities is equal to:    1.00000

the sum of all probabilities is equal to 1 as it should be.

if you draw a sample of 1000 components, then the sum of all probabilities will still be equal to 1 but each possible probability will be much less.
example:
the probability of exactly 106 defective components out of the sample of 1000 components is equal to:
C(1000,106) * .06^106 * .97^894 which is equal to ?????
my calculator is unable to calculate this.
it tells me that C(100,106) is too large for it to handle (overflow error).
it tells me that .06^106 is equal to 0.
it tells me that .94^894 is equal to 9.468934625 * 10^(-25)

suffice it to say that the probability of getting 106 defective components out of a batch of 1000 components is extremely small.

try something like 50 failures out of a batch of 100 components.
you get:
p(50 failures) = C(100,50) * .06^50 * .94^50 which is equal to 3.696656602 * 10^(-34)
that's a very small number.

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you can try to find the line manually or you can use a calculator, like the TI-83 or TI-84 or even the TI-30X-IIS
I used the TI-30X-IIS for this problem.
The calculator told me that the equation for the best fit of this data is:
y = .25x + 50.4
the correlation factor is .693375245 which is a pretty good positive fit.
correlation factor of 1 would be a perfect positive fit.
correlation factor of -1 would be a perfect negative fit.
correlation factor of 0 would be the worst fit (neither negative or positive correlation can be derived).
a table of your data and the derived data from the calculator generated best fit equation is shown below:

x         y         calculator best fit y      manual best fit y
12       54                 53.4                      
14       53                 53.9                 
16       55                 54.4                 
18       54                 54.9                 
20       56                 55.4                 

a picture of your data points and a straight line best fit that i did using the eyeball method gave a pretty close approximation of what the calculator did for me.

you can also manually calculate the best fit linear equation using the method in the following link.
http://www.statisticshowto.com/articles/how-to-find-a-linear-regression-equation/
that, however, is doing it the hard way and definitely not recommended unless you absolutely are required to do so.
calculators are so much easier.
it's a good learning tool, however, for you to appreciate how the calculator does it

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2x + 3y = 18
x = 2y-5
substitute 2y-5 for x in the first equation to get:
2(2y-5) + 3y = 18
simplify by removing parentheses to get:
4y - 10 + 3y = 18
combine like terms to get:
7y - 10 = 18
add 10 to both sides to get:
7y = 28
divide both sides by 7 to get:
y = 4
substitute 4 for y in the second equation of x = 2y-5 to get:
x = 2*4-5 which becomes:
x = 8-5 which becomes:
x = 3
substitute 3 for x and 4 for y in the first equation of 2x + 3y = 18 to get:
2*3 + 3*4 = 18 which becomes:
6 + 12 = 18 which becomes:
18 = 18
values for x and y are good.
x = 3
y = 4

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L = length of rectangle
W = width of rectangle
W = L - 3
P = perimeter = 86 feet.
formula for perimeter:
2L + 2W = 86
substitute L-3 for W to get:
2L + 2(L-3) = 86
simplify by removing parentheses to get:
2L + 2L - 6 = 86
combine like terms to get:
4L - 6 = 86
add 6 to both sides to get:
4L = 92
divide both sides by 4 to get:
L = 23
since W = L - 3, then:
W = 20
formula for perimeter is:
2L + 2W = 86
substitute for L and W to get:
2*23 + 2*20 = 86
simplify to get:
46 + 40 = 86
combine like terms to get:
86 = 86
values for L and W are good.

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tan(t) = sin(t) / cos(t)
sin(t) = 1/csc(t)
cos(t) = 1/sec(t)
by substitution, tan(t) = (1/csc(t)) / (1/sec(t))
this is equivalent to:
tan(t) = (1/csc(t))*sec(t) which is equivalent to:
tan(t) = sec(t)/csc(t)
your original equation is:
sec^2(t)/tan(t) = sec(t)*csc(t)
multiply both sides of this equation by tan(t) to get:
sec^2(t) = sec(t)*csc(t)*tan(t)
substitute sec(t)/csc(t) for tan(t) to get
sec^2(t) = sec(t)*csc(t)*sec(t)/csc(t)
the csc(t) in the numerator and denominator cancel out and you are left with:
sec^2(t) = sec(t)*sec(t)
since sec(t)*sec(t) is equal to sec^2(t), you are done.

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your equation is -3x - 10y = 51
x = (1/3)
y = (-5)
substitute in the equation to get:
-3 * (1/3) - (10 * -5) = 51
simplify to get:
-3/3 - (-50) = 51
simplify further to get:
-1 + 50 = 51
combine like terms to get:
49 = 51
since this is not true, then those values are not solutions to the equation.
if your equation was:
3x - 10y = 51, then:
substituting would get you:
3*(1/3) - (10*(-5)) = 51 which would simplify to:
1 - (-50) = 51 which would simplify further to:
1 + 50 = 51 which would become:
51 = 51
in that case it would be a solution to the equation.
it depends on what your equation is.
the process is the same.
the result is different if i assume your original equation is:
-3x-10y=51 or whether i assume your original equation is:
3x-10y = 51
check your original equation to make sure what it is and you can find the answer depending on that.
the process is the same regardless of the equation.
you replace the variable x with its value and the replace the variable y with its value and then you solve.
the coordinate pair of (1/3, -5) is the (x,y) coordinate pair which means that the value of x is equal to 1/3 and the value of y is -5.

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let L equal the length of the wire and let D equal the diameter of the wire.
let R equal the resistance of the wire.
the formula would be:
R = k * L / D^2
you know that 100 fee of wire with a diameter of .01 inches has a resistance of 90 ohms.
you can use this fact to find k, since k is a constant and will not vary.
your equation of:
R = k * L / D^2 becomes:
90 = k * 100 / (.01)^2
if you multiply both sides of this equation by (.01)^2 / 100 then you get:
90 * (.01)^2 / 100 = k
solve for k to get:
k = .00009
your formula of:
R = k * L / D^2 becomes:
R = .00009 * L / D^2
test this formula out with your original measurements to get:
R = .00009 * 100 / (.01)^2 and you will get:
R = 90 ohms.
use this formula with your new measurements and you will get:
R = .00009 * 68 / (.02)^2 which becomes:
R = 15.3 ohms.