hectictar

To make sure we understand your questions correctly, can you rewrite them to put parenthesees around each numerator and denominator?

For example.... to write the fraction  \(\frac{4a+1}{5b-2}\)  ,  you can write this:  (4a + 1)/(5b - 2)

The way it is currently written, this is your first expression:  \(\frac32y+\frac53y^2\)

But I think that might not be what you meant.

Let's find the equation of the line that passes through  (0, 3)  and  (-8, 0) .

slope  =  [0 - 3] / [-8 - 0]   =   -3 / -8   =   3 / 8

The y-intercept is the y value when  x  is  0 .

We know that the line goes through the point (0, 3) , so when  x  is  0 ,  y is  3 .

The y-intercept is  3 .

The equation of the line in slope-intercept form is...

y  =  (3/8)x + 3

We also know that the line passes through the point  (t, 5) .

So we know it makes a true equation when  x  is  t  and  y  is  5 .

So let's plug in  t  for  x  and  5  for  y , and solve for  t .

5  =  (3/8)t + 3

                              Subtract  3  from both sides of the equation.

2  =  (3/8)t

                              Multiply both sides by  8/3 .

2 * 8/3  =  t

16/3  =  t

t  =  16/3

Here's an edit of terte's graph with the line  y = (3/8)x + 3  and the point  (16/3, 5)  added:

f(x)  =  g(g(x)) - g(x)        Plug in  3  for  x .

f(3)  =  g(g(3)) - g(3)

g(x)  =  2x - 1         Now let's find  g(3) .  Plug in  3  for  x .

g(3)  =  2(3) - 1

g(3)  =  6 - 1

g(3)  =  5

f(3)  =  g(g(3)) - g(3)       Substitute  5  in for  g(3) .

f(3)  =  g(5) - 5

g(x)  =  2x - 1        Now let's find  g(5) .  Plug in  5  for  x .

g(5)  =  2(5) - 1

g(5)  =  10 - 1

g(5)  =  9

f(3)  =  g(5) - 5                Substitute  9  in for  g(5) .

f(3)  =  9 - 5

f(3)  =  4

3x² - 24x + 55

                                                      Factor  3  out of the first two terms.

   =   3(x² - 8x) + 55

                                                      Add  16  and subtract  16  inside the parenthesees.

   =   3(x² - 8x + 16 - 16) + 55

                                                      Factor  x² - 8x + 16  as a perfect square trinomial.

   =   3( (x - 4)² - 16) + 55

                                                      Distribute the  3  to the terms in parenthesees.

   =   3(x - 4)² - 48 + 55

                                                      Combine  -48  and  55  to get  7 .

   =   3(x - 4)² + 7

   =   a(x + d)² + e

So...

a + d + e   =   3 + -4 + 7   =   6

g(x)  =  √x

                       Plug in  16  for  x .

g(16)  =  √16

g(16)  =  4

And...

t(x)  =  3 - g(x)

                                          Since  g(x)  =  √x  , we can substitute  √x  in for  g(x) .

t(x)  =  3 - √x

                                          Plug in  g(16)  for  x .

t(g(16))  =  3 - √[ g(16) ]

                                          Since  g(16)  =  4 , we can substitute  4  in for  g(16) .               

t(g(16))  =  3 - √[ 4 ]

t(g(16))  =  3 - 2

t(g(16))  =  1

Also...

t(g(16))  =  3 - g( g(16) )  =  3 - g( 4 )  =  3 - 2  =  1

The question:

"ABCD is a trapezoid with the measure of base AB twice the measure of the base CD. Point E is the point of intersection of the diagonals. The measure of diagonal AC is 11. Find the length of segment EC. Express your answer as a common fraction."

∠AEB  and  ∠DEC  are vertical angles, so they are congruent.

∠CAB  and  ∠ACD  are alternate angles, so they are congruent.

By the AA similarity theorem, △ABE ~ △CDE, so...

AE / AB   =   CE / CD

The measure of diagonal AC is 11.

AE + CE  =  AC

AE + CE  =  11

AE  =  11 - CE

And the measure of base AB is twice the measure of the base CD.

AB  =  2CD

Now we can use these values of  AE  and  AB .

AE / AB  =  CE / CD

                                                     Substitute  2CD  in for  AB  and  11 - CE  in for  AE.

(11 - CE) / (2CD)  =  CE / CD

                                                     Multiply both sides of the equation by  CD .

(11 - CE) / 2  =  CE

                                                     Multiply both sides of the equation by  2 .

11 - CE   =   2CE

                                                     Add  CE  to both sides.

11  =  3CE

                                                     Divide both sides by  3 .

11 / 3  =  CE

EC  =  11 / 3

Yep! 

This is one way to think of it:

We know...

9 lags  =  20 lugs      Multiply both sides of this equation by  4 .

36 lags  =  80 lugs

And we know...

7 ligs  =  4 lags         Multiply both sides of this equation by  9 .

63 ligs  =  36 lags

63 ligs  =  36 lags       and       36 lags  =  80 lugs    So...

63 ligs  =  80 lugs

This is given in the directions:

"If increasing one variable causes the other to increase, this is called a direct relation.

If increasing one variable causes the other to decrease, then this is called an inverse relation."

If you increase the height of the flagpole, the length of the shadow will increase. Relation is direct.

If you increase the weight of cut, the cost of the cut diamond will increase. Relation is direct.

If you increase the weight of runner, the time to run 2 miles will increase. Relation is direct.

If you increase speed of internet connection, time to load a large file will decrease. Relation is inverse.

(Even in this example, the time to load a large file depends on the speed of internet connection.)

If you increase the HP of a motorcycle engine, the acceleration of cycle will increase. Relation is ___.

If you increase the practice time, the ability to play a guitar will ____. Relation is ___.

Do you know the answer to the last ones?