Calculus

Ans1 : 3x - 2y = 5 (the x and y terms are on the left side while the constant is on the right side).

Ans2 : 2y=3x+5 (divide both side by 2)

y=(3/2)x+(5/2)

this is similar to standard equation of line y = mx + c

When you divide zero by anything or multiple anything by zero, the answer will always be zero.

Well what does y equal? If y equals 7 the 3+7=10

It is equal to 1/4 of a cup.

0

-- 3 miles per second -- 10,800 miles per hour -- 29,030,400 furlongs per fortnight

0

2x + 3 = 9x

Subtract 2x from both sides: 3 = 7x

Divide both sides by 7: 3/7 = x

x = -6 x/-3 + 5 = 7

x/-3 = 2

x = -6

-2x+8=3y
(-2x+8)/3=y
plug in values of x or y into equation to find x or y

To obtain specific values for x or y, you would need 2 equations for the two variables. You can still solve the equation for a specific variable, in terms of the other. For example, to solve for x:
2x + 3y = 8
2x = 8 - 3y
x = (8 - 3y) / 2

3y+2z

Infinity is not a number.

There are different classes of infinity:

The sets of natural numbers, integers, rational numbers all belong to the smallest class, with a cardinality of Aleph-null.

The sets of irrational numbers and real numbers belong to the next higher level of infinity, with cardinality Aleph-One.

Infinity can give rise to a very large number of apparent paradoxes - infinitely many of them?

112

7

x - 14

Solving equations in two unknowns requires two independent equations. Since you have only one equation there is no solution.

The answer is 398.78 cm (approx.). Inches and centimeters are both units of linear measurement. Inches are used in the imperial system whereas centimeters are used in the metric system. To convert from inches to cm, multiply the inch unit by 2.54.

The indefinite integral of cos(sqrt(x))/sqrt(x) dx

This is an integral that can be solved by simple u-substitution. This integral in and of itself cannot be solved by any simple antiderivative pattern, but by using u-substitution we can get this integral to resemble something more familiar.

If we set a dummy variable "u" equal to sqrt(x), we could get the integral to look like cos(u). Although this substitution causes the sqrt(x) to disappear inside the cosine function, the variable of integration (dx) is still in terms of x. However, since:

u=sqrt(x)

du=d/dx of sqrt(x) [the derivative of sqrt(x)]

du=(1/2)x(1/2)-1 dx=(1/2)x-1/2 dx=1/(2sqrt(x)) dx

This implies that the 1/(2sqrt(x)) inside of the integral can be replaced by du. All that exists inside of the integral is 1/sqrt(x) though, but by dividing the 1/(sqrt(x)) inside of the integral by the needed 1/(2sqrt(x)), we get that multiplying the entire integral by a constant of 2 would let us solve this integral. So the integral above that was unsolvable in terms of x is now solvable in terms of u:

2 times the integral of cos(u) du

If any limits of integration existed, you could put them through the relationship u=sqrt(x), you can change the limits into terms of u as well. If, for instance, the integral was from x=3 to x=7, the integral in terms of u would be from u=sqrt(3) to u=sqrt(7).

So, the antiderivative of cos(u) would be sin(u). This is a basic antiderivative pattern, one that you should have memorized. So, since the integral in terms of u was multiplied by a constant of 2, the final answer should be as well. So, we get the antiderivative to be:

2sin(u)+C [the "+C" exists since this is an indefinite integral with no limits of integration]

By resubstituting u=sqrt(x) back in, we get the final antiderivative answer in terms of x to be:

2sin(sqrt(x))+C

BUT: realize that if there were limits of integration, you must substitute them into the integral in terms of whatever variable the antiderivative is in. If you were to use the above antiderivative in terms of x, you would have to use limits of integration that were in terms of x. Likewise, if you were to use the antiderivative while it was still in terms of u, you could solve for it at that point by simply using the limits in terms of u as we discussed earlier.

x2 + x2 = 2x2. Don't be thrown by squares (or other powers).

y + y = 2y. Just treat x2 like you treat y. But remember, you can only add together the same powers of x, you cannot add together x and x2 (or other powers of x).

6

It is a trial solution, as mentioned in the question!

3

MgCl2 + N2 = Mg3N2 + 3 Cl2