Solving equations 1 / M + 1 / N = 1 / 12 8 / M-3 / N = 3 / 10 1/m+1/n=1/12 　　　　　　　8/m-3/n=3/10

Let x = 1 / m, y = 1 / n
x+y=1/12
8x-3y=3/10
3x+3y=1/4
Two formula addition
11x=3/10+1/4
11x=11/20
x=1/20
y=1/12-x=1/12-1/20=1/30
m=1/x=20
n=1/y=30
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To solve the equations: M + N3 + n − M4 = − 14m + 86 − 5 (n + 1) 12 = 2

The original equation system can be reduced to 2m + 14N = − 62m − 5N = 13 (1) - (2) to get n = - 1, and substituting (2) to get m = 4. Therefore, the solution of the original equation system is m = 4N = − 1

It is known that the solution of the system of equations 2 х + my = 4x + 4Y = 8 is a positive integer solution. The value of integer m and the solution of the system of equations are obtained

2x+my=4………… （1）
x+4y=8……………… （2）
From (2), it is concluded that:
x=8-4y
If the solution is a positive integer, then 8-4y > 0
y

M ^ 2 + 2Mn + 2n ^ 2-6n + 9 = 0 (3m + n) ^ 2 (n-3) ^ 2, so n = 3, M = - 3, so m of n ^ 2 = - 1 of 3
(1) If x ^ 2 + 4x + 4 + y ^ 2-8y + 16 = 0, find the value of Y (2) if x ^ 2 + 2Y ^ 2-2xy + 2Y + 1 = 0, find the value of X + 2Y

(1) The result is - 2. X ^ 2 + 4x + 4 + y ^ 2-8y + 16 = (x + 2) ^ 2 + (y-4) ^ 2 = 0
(2) X = y = - 1, so the result is - 3.. x ^ 2 + 2Y ^ 2-2xy + 2Y + 1 = (X-Y) ^ 2 + (y + 1) ^ 2 = 0

Square of factorization factor M - square of factorization factor N + 5m + 5N

=(m+n)(m-n)+5(m+n)
=(m+n)(m-n+5)

Prove that the square of a is equal to m square minus n square, B is equal to m square plus N square, C is equal to 2Mn, and N is the three sides of a right triangle

b²-a²
=(m²+n²)²-(m²-n²)²
=(m²+n²+m²-n²)(m²+n²-m²+n²)
=4m²n²
=(2mn)²
=c²
so
b²=a²+c²
From the inverse theorem of Pythagorean theorem
ABC is a right triangle

The square of 3 = 4 + 5, and the square of 5 = 12 + 13. This is not a coincidence, but there are rules to find. What are the rules? Please use Pythagorean theorem to prove it

1、 Description of law: if the square of an integer (a) can be expressed as the sum of two continuous positive integers (B, B + 1), then the sum of the square of the number (a) and the smaller positive integer (b) is equal to the square of the larger positive integer (b + 1)
a²+b²=(b+1)²
Where a & sup2; = B + (B + 1),
2、 Proof of the law:
Since a & sup2; = B + (B + 1) = 2B + 1, then
a²+b²
=(2b+1)+b²
=(b+1)²
The law has been proved

How to prove Pythagorean theorem? What is its principle,
I remember that one of the supplementary exercises on the eighth day of the eighth day proved the Pythagorean theorem, but it was taken away by the teacher. Who remembers to tell me! The most important thing is what is the key to prove the Pythagorean theorem, and what kind of thought, it's OK to make a topic,

There are many methods, but the principle is the same: Area unchanged! If you don't understand, you can continue to ask

If the square of M minus 4m is less than or equal to 0, the value range of M can be obtained

The square of x minus 2 (M + 1) x + m + 3 = 0

B * 2-4ac should be greater than or equal to 0
So [2 (M + 1)] * 2-4 (M + 3)
=4m*2+4m-8
=4 (M + 1 / 2) * 2-9 should be greater than or equal to 0
Let 4 (M + 1 / 2) * 2-9 = 0, then M = 1 or M = - 2
So m is greater than or equal to 1 or m is less than or equal to - 2

Solving equations 1 / M + 1 / N = 1 / 12 8 / M-3 / N = 3 / 10 1/m+1/n=1/12 8/m-3/n=3/10