Question

Числа а ,b,c и d таковы ,что а + b=c+ d не равно нулю, а c=bd.Докажите, что а +с=b+d

Answer 1

Решение смотри на фото

Answer 2

Given Information

We are given that there are four numbers, a, b, c, and d, such that a + b = c + d is not equal to zero, and c = bd.

Proof

To prove that a + c = b + d, we can start by substituting c with bd in the equation a + b = c + d.

a + b = c + d

Substituting c with bd:

a + b = bd + d

Now, let's try to express a and b in terms of d.

From the equation c = bd, we can solve for b:

c = bd

Dividing both sides by d:

b = c/d

Similarly, we can solve for a:

a = c - b

Substituting the values of a and b in the equation a + b = bd + d:

c - b + b = bd + d

Simplifying:

c = bd + d

Factoring out d:

c = d(b + 1)

Now, we can substitute this expression for c in the equation a + c = b + d:

a + d(b + 1) = b + d

Expanding the expression:

a + bd + d = b + d

Rearranging the terms:

a + bd = b

Substituting the value of a from the equation a = c - b:

c - b + bd = b

Rearranging the terms:

c + bd = b + b

Simplifying:

c + bd = 2b

Dividing both sides by 2:

(c + bd)/2 = b

Now, we can substitute this expression for b in the equation a + bd = b:

a + bd = (c + bd)/2

Multiplying both sides by 2:

2a + 2bd = c + bd

Subtracting bd from both sides:

2a + bd = c

Now, we can substitute the value of c from the equation c = bd:

2a + bd = bd

Subtracting bd from both sides:

2a = 0

Dividing both sides by 2:

a = 0

Since a = 0, we can substitute this value in the equation a + bd = b:

0 + bd = b

Simplifying:

bd = b

Dividing both sides by b:

d = 1

Now, we can substitute the values of a and d in the equation a + c = b + d:

0 + c = b + 1

Simplifying:

c = b + 1

Therefore, we have proved that a + c = b + d.

Note: The proof assumes that a + b = c + d is not equal to zero, as stated in the given information. If a + b = c + d is equal to zero, the proof does not hold.