Calculus

-13.9x = 5.7-12.5x ( +12.5x)

-1.4x = 5.7 (/-1.4)

x = 5.7/-1.4 = -4.07 to 2 decimal points

6x+7y=-6 3(-2x-3y=2)

+________

-2y=0 => y=0

Therefore x=-1

354678 x 34562 = 12258381036

Plug in various values of x and solve for y. [This can be done for any value of x.]

For example, for x=0,

2*0+3y=15

3y=15

y=15/3=5

So (0,5) is one coordinate.

For x=1,

2*1+3y=15

2+3y=15

3y=13

y=13/3

So (1,13/3) is another coordinate.

For x=2,

2*2+3y=15

4+3y=15

3y=11

y=11/3

So (2,11/3) is another coordinate.

Grammar in mathematics is important so that the question cannot be misunderstood.

The question is asking for the solution to 1/(x + 8) + 6/(x - 8).

Where fractions are being added or subtracted then the denominators need converting to a common denominator. With numbers this new denominator would be the Lowest Common Multiple. With algebraic terms then simply take the product.

The common multiple of (x + 8) and (x - 8) is (x + 8)(x - 8) = x2 - 82

1/(x + 8) = 1 x (x - 8)/x2 - 82 = x - 8/x2 - 82

6/(x - 8) = 6 x (x + 8)/x2 - 82 = 6x + 48/x2 - 82

1/(x + 8) + 6/(x - 8) = x - 8/x2 - 82 + 6x + 48/x2 - 82 = 7x + 40/x2 - 82

A solution means: Show the numerical values of 'x' and 'y'.

You can't find either one with only a single equation. You need another one.

When you have two equations, you can find values for both 'x' and 'y'.

27, 54, 81, 108, 135, 162, 189, 216, 243, 270, 297, 324, 351, 378, 405, 432, 459, 486, 513, 540, 567, 594, 621, 648, 675 . . . ∞

cosx√y=5

Right away you can tell that there'll be a domain error because if you were to take cosine inverse five equals X root y then there would be an error

6.37 = 2.62+X


Then subtract 2.62 from 6.37


6.37 = 2.62+X
-2.62 -2.62


Therefore you end up with





3.75 = X

There is no single value that this equals. There could be any number of answers to it depending on the value of x.

If you want to know what x equals when a specific value is assigned to the equation, you can do this by solving the equation for a 0 value, and then factoring the result.

For example, let's say you you want to know the value of x when the equation equals negative four:

x2 + 8x + 12 = -4
∴ x2 + 8x + 16 = 0
∴ (x + 4)2 = 0
Now you simply plug in the value on the left side of the equation that would give you a result of 0. In this case, it would be negative four, as negative 4 plus 4 equals 0, which is also 0 when squared.
∴ x = -4

Similarly, if you say that the equation comes to a value of 5:
x2 + 8x + 12 = 5
∴ x2 + 8x +7 = 0
∴ (x + 7)(x + 1) = 0
In this case, there are two possible outcomes: negative 7 and negative 1. That's because either one of those will produce a zero value for this equation.
∴ x = -7 or x = -1


Alternately, what happens if the equation comes to a value of 1?

x2 + 8x + 12 = 1
∴x2 + 8x + 11 = 0

In this case, there is no simple way to solve it by factoring, so instead you can plug it into the quadratic formula, x = [-b +/- √(b2 - 4ac)] / 2a

∴ x = [-8 +/- √(82 - 4×1×11)] / (2×1)
∴ x = [-8 +/- √(64 - 44)] / 2
∴ x = (-8 +/- √20) / 2
∴ x = -4+√20, or -4-√20
∴ x ≈ 0.47214, or x ≈ -8.42714

As shown, there can be any number of correct answers to this, as there is no specified value for x. Once a value is specified, then it will always have at most two answers, and at least one.

if, for example, 0=x^2+5x+6, then:

0=(x+3)(x+2)

the numbers '3' and '2' must add to give you 5, and multiply to give you 6.

SO:

0=x^2+(a+b)x+(ab)x

0=(x+a)(x+b)

(10,13)

13.935

7

Subtract 4 from both sides and add y to both sides. This would give you 2y=14.

Then, divide both sides by 2. y=7.

52,560,000

The way you can get that number is simple:

10 x 0.10 = 1

1 x 60 = 60

60 x 24 = 1,440

1440 x 365 = 525,600

525,600 x 100 = 52,560,000

You see, you start out by multiplying the number of numbers you can count every 10 seconds, you multiple that number by the number of minutes in a hour, you multiple that number by the number of hours in a day, you multiple that number by the number of days in a year, and you multiple that number by 100 years.

So, if you constantly counted by 1 every 1 second, for 100 years, the number you would end up with is 52 billion, 560 hundred thousand.

24

Given:

a = 2πr2 + 2πrh

h = 113/r2

∴ a = 2πr2 + 2πr(113/r2)

∴ a = 2πr2 + 226π / r

∴ da/dr = 4πr + 226π ln r

a=11