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So for this question you can just cross multiply! 

\((1+\frac{1}{x}) \cdot x=1\)

\(x+1=1\)

\(x=0\)

but, this isn't possible since 0 cannot be a denominator, which means that there is no solution.

slope  =  (y₂ - y₁) / (x₂ - x₁)

 -3/2  =  ( b - 0 ) / ( 0 - 8 )

 -3/2  =  b / -8                        (multiply both  sides by -8)

   12  =  b

Lizzie's rate is 1 fence in 6 hours, or 1/6.

Alex's rate is 1 fence in 8 hours, or 1/8.

Since they work the same amount of time, time = t.

amount completed = rate x time

amount Lizzie does  +  amount Alex does  =  total amount completed

                    (1/6)(t)  +  (1/8)(t)                    =  1 fence painted

                           (7/24)(t)                            =  1

                                     t                              =  1 / (7/24) 

                                                                 t  =  24/7 hours     

9^2x  =  (3²)^2x  =  3^4x 

Since  9^2x  =  3^{3x + 1}   --->   3^4x  =    3^{3x + 1}    --->   4x  =  3x + 1   --->   x  =  1

Let  A = (3,0)     B = (0,3)     C = (9,0)     D = (0,9)     O = (0,0)

This is a trapezoid with AB and CD being the parallel sides (the bases).

The formula for the area of a trapezoid:  A = ½·h·(b₁ + b₂)

b₁ = side AB     b₂ = side CD

Since triangle(OAB) is 45^o-45^o-90^o right triangle with OA = 3 and OB = 3,, AB = 3·sqrt(2).

Since triangle(OCD) is 45^o-45^o-90^o right triangle with OC = 9 and OD = 9,, CD = 9·sqrt(2).

Now to find the height --  draw line segment OXY perpendicular to both AB and CD, with

X the point of intersection with AB and Y the point of intersection with CD.

Since triangle(OAX) is 45^o-45^o-90^o right triangle with OA = 3, OX = 3/sqrt(2).

Since triangle(OCY) is 45^o-45^o-90^o right triangle with OC = 9, OY = 9/sqrt(2).

XY = OY - OX = 9/sqrt(2) - 3/sqrt(2) = 6/sqrt(2) or 3·sqrt(2).

This is the height of the trapezoid.

A = ½·h·(b₁ + b₂​) = ½·[ 3·sqrt(2) ]·[ 3·sqrt(2) + 9·sqrt(2) ]

                             = ½·[ 3·sqrt(2) ]·[ 12·sqrt(2) ]

                             = 36

If  a = 1,  b = 100

    a = 2,  b = 25

    a = 5,  b = 4

    a = 10,  b = 1

The circle with center A has a radius of 10.

The circle with center B has a radius of 7.

These two circles are externally tangent; therefore, AB = 17.

There is an external tangent to these two circles. Both circles lie on the same side of this external tangent.

Label this external tangent XYA, where X is the point of tangency to circle A and Y is the point of tangency 

to circle B. A is the point where this line intersects line AB.

Let x represent the length BC.

Triangle(AXC) is similar to triangle(BYC) by AA.

We have:  AX / AC  =  BY / BC     --->     10 / (17 + x)  =  7 / x

              cross-multiply:                                         10x  =  7(17 + x)

                                                                              10x  =  119 + 7x

                                                                                3x  =  119

                                                                                  x  =  119/3

anyone?

100032 = (1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 192, 521, 1042, 1563, 2084, 3126, 4168, 6252, 8336, 12504, 16672, 25008, 33344, 50016, 100032) > 28 (divisors)

By similar triangles, BC = 331/10.

sumfor(n, 101, 200, n^2)=2,348,350

Like @NoWorkNeeded said, if he missed both times we multiply 0.4 by 0.4 to get 0.16.

As a fraction, this is 4/25

Ok, 

We apply the rule where a/b = c/d is ad = bc. 

3 . 2x = x+10 . 1

6x = x+10

We isolate 6x by subtracting x from both sides.

5x = 10

x = 2

I think you mean when they are 180 degrees apart:

180=abs((30*1) - (5.5*M)), solve for M

M =420 / 11 =38.181818........minutes - or 38 minutes and about 11 seconds after 1 O'clock.

Ax + y  =  B     --->     y  =  B - Ax

x + By  =  A     --->     x + B(B - Ax)  =  A                                (substitute)

                                   x + B² - ABx  =  A                                (distribute)

                                           x - ABx  =  A - B²                         (subtract)

                                         x(1 - AB)  =  A - B²                        (factor)

                                                     x  =  (A - B² / (1 - AB)        (divide)

A gutter can hold up to 20 liters of rain water, (the gutter is a slip in half cylinder) and the diameter of it is 14 cm. what is the length of the gutter in meters?

a + b  =  14     --->     a  =  14 - b

a³ + b³  =  812      --->     (14 - b)³ + b³  =  812           (substitution)

   [ 2744 - 3(14²)b + 3(14)b² - b³ ] + b³  =  812           (cubic expansion)  

                              2744 - 588b + 42b²  =  812

                              42b² - 588b + 1932  =  0

                                21b² - 294b + 966  =  0

Now use the quadratic formula ...

can you help me with loci

I don't know how to post a diagram. Sorry.

To bisect a line segment:

-- open you compass to a little more than half of the length of the line segment.

-- draw an arc from one of the endpoints through the line segment.

-- draw an arc from the other endpoint through the line segment.

-- there should be two points of intersection of the arcs, one on one side of the line segment;

          the other on the other side.

-- connect these two points of intersection with a line

   -- this line is the perpendicular bisector of the original line segment.

   -- where this line crosses the original line segment is the midpoint of that line segment.