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Plug the given data:
m=1/3; (6,3)
into the "point-slope" form:
y - y1 = m(x - x1)
y - 3 = (1/3)(x - 6)
y - 3 = (1/3)x - 2
y = (1/3)x + 1
the above is now in "slope-intercept" form

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Andrew can paint a room by himself in 8 hours. With the help of Mike, it will only take him 3 hours. How long does it take Mike by him self?
.
Let x = time (hours) it takes for Mike by himself
then
3(1/8 + 1/x) = 1
multiplying both sides by 8x:
3(x + 8) = 8x
3x + 24 = 8x
24 = 5x
24/5 = x
4.8 hours = x

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Find the area of an isosceles triangle with legs of 12cm and a base of 6cm.
.
area of any triangle = (1/2)bh
where
b is base
h is height
.
the problem gives you the base (b) as 6 cm.
You need to find the height (h). The height (h) splits the isosceles triangle in half and it is a right triangle -- use Pythagorean theorem to find h:
3^2 + h^2 = 12^2
9 + h^2 = 144
h^2 = 135
h = sqrt(135)
h = sqrt(5*3*3*3)
h = 9*sqrt(15)
.
area = (1/2)bh
area = (1/2)(6)(9*sqrt(15))
area = (3)(9*sqrt(15))
area = 27*sqrt(15) sq cm
area = 104.57 sq cm

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ln(2-2x)=ln(x+5)+ln(x+4)
ln(2-2x)=ln(x+5)(x+4)
(2-2x)=(x+5)(x+4)
2-2x= x^2 +9x + 20
2 = x^2 + 11x + 20
0 = x^2 + 11x + 18
0 = (x+9)(x+2)
x = {-9,-2}
Testing each of the solution above, we see that -9 is an "extraneous solution" and must be thrown out leaving:
x = -2

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if you are traveling 79mph how far would yougo in 0.86sec.
.
convert 0.86 sec to hours:
0.86sec * 1min/60sec * 1hr/60min
.000239 hours
.
since d=rt
d = 79*.000239
d = 0.0189 miles
or
d = 99.65 feet

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if it take 4 hours to fill a pool with the drain closed and 5 hours to empty a pool with the drain open, how long does it take to fill a pool with the drain open.
.
Let x = time (hours) it takes to fill pool w/drain open
then
x(1/4 - 1/5) = 1
multiplying both sides by 20:
x(5 - 4) = 20
x(1) = 20
x = 20 hours

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the total area of a sphere is 144 pie square centimeters. what is the diameter?
.
since the area of a sphere is given as:
4(pi)r^2
.
We set the above equal to 144 to find the radius:
4(pi)r^2 = 144(pi)
dividing both sides by pi:
4r^2 = 144
dividing both sides by 4:
r^2 = 36
take the square root of both sides:
r = 6 cm
.
But, the problem wanted the diameter...
diameter = 2r
diameter = 2(6)
diameter = 12 cm (answer)

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A 15 m pole is leaning against a wall. The foot of the pole is 10 m from the wall. What is the angle that the pole makes with the ground?
.
Apply Trigonometry...
Let x = angle
then
cos x = 10/15
x = cos^(-1)(10/15)
x = cos^(-1)(2/3)
x = 48.2 degrees

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If a right triangle has legs of 6 cm and 8 cm, then the hypotenuse is ?
Apply Pythagorean theorem:
Let h = hypotenuse
then
6^2 + 8^2 = h^2
36 + 64 = h^2
100 = h^2
10 cm = h

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log3(c - 4) + log3(c + 6) = 4
log3(c - 4)(c + 6) = 4
(c - 4)(c + 6) = 3^4
c^2 + 6c - 4c -24 = 81
c^2 + 2c -105 = 0
apply the quadratic formula to find the roots:
c = {9.3, -11.3}
throw out the negative root leaving:
c = 9.3
.
details of quadratic to follow:

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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=424 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 9.295630140987, -11.295630140987. Here's your graph:

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y = 6 x + 7 - x^2
rearranging:
y = -x^2 + 6x + 7
.
The x-coordinate of the vertex is at:
x = -b/(2a) = -6/(2*(-1)) = -6/(-2) = 3
.
The y-coordinate:
y = -x^2 + 6x + 7
y = -3^2 + 6(3) + 7
y = -9 + 18 + 7
y = 16
.
Answer: (3, 16)

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Next time, you should write your expression as:
5^(x+3) = 2(x-6)
log(5) * (x+3) = log(2) * (x-6)
xlog(5) + 3log(5) = xlog(2) - 6log(2)
xlog(5) - xlog(2) = -6log(2) - 3log(5)
x(log(5) - log(2)) = -6log(2) - 3log(5)
x = (-6log(2) - 3log(5)) / (log(5) - log(2))
x = (-3.9) / (.40)
x = -9.76

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3/8 + 3/5
change both terms so they have a common denominator of 40:
15/40 + 24/40
(15 + 24)/40
39/40

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Find two consecutive negative integers whose product is 90
.
Let x = first negative answer
then
x+1 = second negative answer
x(x+1) = 90
x^2 + x = 90
x^2 + x - 90 = 0
(x+10)(x-9) = 0
x = (-10, 9)
Since we're looking for a negative answer, throw out the positive answer leaving:
x = -10 (first negative number)
.
Second negative number"
x+1 = -10+1 = -9
.
Answer: -10 and -9

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an element decays in such a way that every 50 years the amount of the element of the element has decreased by 15% in the year 1900,120mg of element was present.
a)what is amount in 2000
.
A = Pe^(rt)
where
A is amount after time t
P is the initial amount
r is the rate of growth/decline
t is time
.
first, we determine r...
Let x = initial amount
then
from: "an element decays in such a way that every 50 years the amount of the element of the element has decreased by 15%"
x - .15x = xe^(50r)
x(1 - .15) = xe^(50r)
(1 - .15) = e^(50r)
.85 = e^(50r)
ln(.85) = 50r
ln(.85)/50 = r
-0.0033 = r
.
Our formula then is:
A = Pe^(-0.0033t)
.
Now, we can answer the question:
in the year 1900,120mg of element was present.
a)what is amount in 2000
t = 2000-1900 = 100
A = 120e^(-0.0033*100)
A = 120e^(-0.33)
A = 86.7 mg

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Algebraic expression:
1.Eight less than e
8 - e
2.Five subtracted from t
t - 5
3.The sum of x and y
x + y
4.one more than a number p
p + 1
5.The product of v and 4 times i
v(4i)
or
4iv

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A circle has a radius of 5 cm ..what is the length of an arc measuring 80 degrees
.
circumference of a circle = d(pi)
circumference of a circle = 2r(pi)
circumference of a circle = 2(5)(3.14)
circumference of a circle = 31.4 cm
.
Since there is 360 deg in a circle:
length of the arc must be:
80/360 * 31.4 = 6.98 cm

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a right triangle has a leg that measures 5 meters, the angle at the intersection of that same leg and the hypoteneuese is 35 degrees.... How do you find the measure of the hypoteneuese?
.
You apply trigonometry.
cos = adjacent/hypotenuse
.
Let h = hypotenuse
then
cos 35 = 5/h
h*cos 35 = 5
h = 5/cos 35
h = 6.10 meters

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t^2-8t+15=0
factoring the left:
(t-3)(t-5) = 0
t = {3, 5}

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Create a table of data:
year time
2000 1
2005 5
2010 10
.
Plot the above three points on graph paper.
Because the dots appears to be in a straight line, you can assume it is linear.
.
Determine slope:
m = (y2-y1)/(x2-x1) using first and last points
m = (10-1)/(2010-2000)
m = (9)/(10)
m = 9/10
.
using the slope above and one point (2000,1)
plug it into the "point-slope" form:
y - y1 = m(x - x1)
y - 1 = (9/10)(x - 2000)
y - 1 = (9/10)x - (9/10)2000
y - 1 = (9/10)x - (9)200
y - 1 = (9/10)x - 1800
y = (9/10)x - 1801 (this is your equation)
.
For, 2015:
y = (9/10)x - 1801
y = (9/10)2015 - 1801
y = (9)201.5 - 1801
y = (9)201.5 - 1801
y = 1813.5 - 1801
y = 12.5 minutes

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(8,7) and (9,-9)
m (slope) = (y2 - y1)/(x2 - x1)
m (slope) = (-9 - 7)/(9 - 8)
m (slope) = (-16)/(1)
m (slope) = -16

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Set x equal to zero to find the y-intercept:
y=-9x^2+5x-7
y=-9(0)^2+5(0)-7
y=-7
So, your y-coordinate is -7

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I'm assuming the 't' is part of the exponent... if not, write back.
2500=4e^(58t)
divide both sides by 4:
625 = e^(58t)
take the natural log (ln) of both sides:
ln(625) = 58t
divide both sides by 58:
ln(625)/58 = t
.111 =t

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One number is 4 more than 3/8 of another number. Total of both is 427.
.
Let x = one number
and y = another number
.
From: "One number is 4 more than 3/8 of another number."
x = 4+ (3/8)y (equation 1)
.
From: "Total of both is 427"
x + y = 427 (equation 2)
.
Solve equation 2 for y:
x + y = 427
y = 427-x
.
Substitute above into equation 1 to find x:
x = 4+ (3/8)y
x = 4+ (3/8)(427-x)
8x = 32+ (3)(427-x)
8x = 32+ 1281-3x
11x = 32+ 1281
11x = 1313
x = 119.36
.
Substitute above into equation 2 to find y:
x + y = 427
119.36 + y = 427
y = 307.64

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16x to the second power-16x+4for x=2
Given:

then they say x=2
substitute it in:

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The line y=2x-k passes through the point (4,0)
.
Since the line passes through (4,0)
this means that
x = 4
y = 0
.
plug it in to:
y = 2x-k
0 = 2(4)-k
0 = 8-k
adding 'k' to both sides:
k = 8

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when applying the "quadratic formula"
Instead of:

It should be:

.

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Find equation of line passing through the point (5,-2) and (-3,4)
.
First, find the slope:
m = (y2-y1)/(x2-x1)
m = (4-(-2))/(-3-5)
m = (4+2)/(-3-5)
m = (6)/(-8)
m = -3/4
.
Next, use one of the two given points (5, -2) and the slope above -- plug it into the "point-slope" form:
y - y1 = m(x - x1)
y - (-2) = (-3/4)(x - 5)
y + 2 = (-3/4)x + 15/4
y = (-3/4)x + 15/4 - 2
y = (-3/4)x + 15/4 - 8/4
y = (-3/4)x + 7/4
.
So, YES, you are correct!

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A stone is thrown directly upward from a height of 30ft with a velocity of 60ft/sec. The height of the stone t seconds after it has been thrown is given by the function s(t)= - 16t^2 + 16t + 30. Find the time at which the stone reaches its maximum height and find that maximum height.
.
Time reaching maximum height is at the vertex:
t = -b/(2a)
t = -16/(2*(-16))
t = -16/(-32)
t = 1/2 seconds
.
Maximum height:
s(1/2)= - 16(1/2)^2 + 16(1/2) + 30
s(1/2)= - 16(1/4) + 8 + 30
s(1/2)= - 4 + 8 + 30
s(1/2)= 34 feet

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the difference between the tens digit and the unit digit of a two digit number is 3. Suppose the tens digit is one less than twice the units digit. what is the number?
.
Let x = tens digit
and y = units digit
then
from: "the difference between the tens digit and the unit digit of a two digit number is 3"
x-y = 3 (equation 1)
from: "Suppose the tens digit is one less than twice the units digit"
x = 2y-1 (equation 2)
.
Start with equation 1:
x-y = 3
substitute definition of x from equation 2:
2y-1-y = 3
y-1 = 3
y = 4
.
Substitute above into equation 1 to find x:
x-y = 3
x-4 = 3
x = 7
.
Solution: the number is 74

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Two angles are complementary. they have measures of (7x+2)degree and (3x-2)degree, respectively. what is the value of x?
.
Since the sum of complementary angles is 90 degrees we get:
(7x+2) + (3x-2) = 90
7x+2 + 3x-2 = 90
10x+2-2 = 90
10x = 90
x = 9