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We plug the values into the quadratic equation to get the solutions \(\frac{11\pm\sqrt{41}}{10}\). I think you can do the rest from there.

To find the y-intercept, we plug 0 into the given equation for x.

3x-7y=21

3(0)-7y=21

-7y=21

y=-3

The y intercept is (0, -3).

To find the x-intercept, we plug 0 into the given equation for y.

3x-7y=21

3x-7(0)=21

3x=21

x=7

The x-intercept is (7, 0).

Plugging these values into the distance formula, the answer is \(\sqrt{58}\).

Use the quadratic formula to find a and b.

Use these values to find the answer.

Have you found a and b?

A horizontal line is the in the form of y=___. Then the line passing through (-5, 8) must be y=8. A vertical line is in the form of x=___. The the line passing through (4, -1) must be x=4. The point of intersection is therefore (4, 8).

The maximum product will occur at one-half the sum.

One-half of 246  =  123:  123 x 123  =  15129.

If you want to graph this, use the equation:  y  =  x(246 - x)

Its vertex occurs at (123, 15129).

Notice that the white shapes are 1/6 of a circle. There are 6 of them so in total, their area is equivalent to one circle. We need to find the radius, which is 1/2 of the side of the equilateral triangle. So the radius of the circle is 1/2*10=5. Then the area of the circle is \(\pi{r}^2=\pi(5)^2=25\pi\). The shaded area is then the equilateral triangle x 2 minus 25pi.

To find the area of the equilateral triangle, we already have the base but we need the height. To find this we use the pythagorean theorem. We are using half of the equilateral triangle in this step.

a^2+b^2=c^2

5^2+b^2=10^2

25+b^2=100

b^2=75

b=\(5\sqrt{3}\)

So the area of the equilateral triangle is

 \(\frac{1}{2}\cdot5\sqrt{3}\cdot10\\ =5\cdot5\sqrt{3}\\ =25\sqrt{3}\)

Multiply this by 2 to get \(50\sqrt{3}\). Now we subtract 25pi. Therefore the area of the shaded area is \(50\sqrt{3}-25\pi\).

2xy + x + 2y  =  899

I re-wrote it:     2xy + x + 2y  =  899     --->     2xy + 2y =  899 - x     --->     (2x + 2)y  =  899 - x

         --->     y  =  (899 - x) / (2x + 2)

From this, I could get these solution;  (11,37)     (19,22)     (35,12)     (59,7)     

                                                            (99,4)       (112,3)     (179,2)     (299,1)

We write the equation of this line in slope intercept first. The slope is -2 and the y-intercept -3 so the equation is y=-2x-3. Then we convert this into standard form:

y=-2x-3

2x+y=-3

Write x+5y=7 in slope intercept form.

x+5y=7

5y=-x+7

y=-x/5+7/5

The slope of the perpendicular line, 15x+Py=Q must be 5.

15x+Py=Q

Py=-15x+Q

y=-15/Px+Q/P

-15/P must equal to 5 so P=-3. We substitute the given point into 15x-3y=Q.

15x-3y=Q

15(-8)-3(3)=Q

-120-9=Q

Q=-129

(-3, -129)

Case 1:  3x - 7 >= 0     and     x + 1 >= 0

                    3x >= 7                      x >= -1

                      x >= 7/3

               Comgining the above:  x >= 7/3

               3x - 7  =  x + 1

                    2x  =  8

                      x  =  4

Case 2:  3x - 7 >= 0     and    x + 1 < 0

                    3x >= 7                    x < -1

                      x >= 7/3

               Combining the above:  impossible

Case 3:  3x - 7 < 0     and     x + 1 >=0

                    3x < 7                     x >= -1

                     x < 7/3

               Combining the above:  -1 <= x < 7/3

                -(3x - 7) = x + 1

                  -3x + 7 = x + 1

                         -4x = -6

                            x = 3/2

Case 4:  3x - 7 < 0     and     x + 1 < 0

                    3x < 7                      x < -1

                     x < 7/3

              Combining the above:  -1 < x < 7/3

              -(3x - 7) = -(x + 1)

                -3x + 7 = -x - 1

                        -4x = -11

                           x = 11/4     (Impossible, not in the required region for x)

The two answers are in bold.

Here's a hint: use exponential notation.

n = 0     should work  

Put in x = 0     calculate y    this is the   y -axis intercept   (0,y)

Put in y = 0     calculate x    this is the   x axis intercept     (x,0)

Then use distance formula to calculate the distance between the two points ...... 

so we do 234/18=13 so the answer is 13 since the area of this rectangle=18*x=234 so we find x=13 so the answer is 13

Yeah I agree

You're welcome ! 

say the sides of the tank are 8    (for ease)

tank one sphere volume = 4/3 pi r^3    = 4/3 pi 8                        = 4/3 pi 2^3

tank 2                                 4/3 pi r^3  =   4/3 pi 1^3  x 8 spheres = 4/3 pi 2^3

tank3                                  4/3 pi r^3  = 4/3 pi 1/2^3  x 64 sphere=  4/3 pi 2^3  

Looks like they will all hold the same amount of water !!!!

Thank you.

thp = 1000x - x

This is just flat rate taxation.    rate .001   or  .1 %

There is something incorrect in your question.....as long as I have greater income I will have greater take home pay.....so there is no limit (or answer) on take home pay.....    what is the taxation again?

100 000       would only have   100 tax

1000 000   would only have  1000  tax     

Higher incomes will always have higher THP

Thank you.

divide x-2   into   x-1  :

       ___1_1/x-2_________

x-2  |  x-1

          x-2       

              1

how did you do that?

It looks like it IS simplified....I suppose you could divide it out to get    1  1/(x-2)   

simplify it

What would you like to do with that EXPRESSION ?