This will be useful later!

Algebra Level 1

There are 4 numbers, a, b, c & d, such that ac = bd and ab = cd

If these conditions are fufilled, is it always true that ad = bc?

In some cases, but not always. Yes, always No, never

1 solution

Mary Arans
Aug 12, 2018

There is an easy solution: if ac = bd, a = bd/c. c = ab/d = bd/c × b/d = b^2/c b^2 = c^2, so b = c. Using this fact, ab = db, so a = d. If a = d and b = c, a^2 = c^2, so a = c. Since a = c, ab = ad so b = d. Taking these into account means a = b = c = d, so ad = bc.

But what if we started with $a = 0, d = 0$ , and $b \ne 0$ and $c \ne 0$ then we would have $ac = bd = 0$ , and $ab = c d = 0$ , however, $0 = a d \ne bc$ . As another example, suppose $a = d = 3$ , and $b = c = 1$ , then $a c = b d = 3$ and $a b = c d = 3$ , however $a d = 9$ , and $b c = 1$ , so $a d \ne bc$ .

Hosam Hajjir - 2 years, 10 months ago

I agree with you. Maybe you can post a report about this. (I actually posted one already.)

X X - 2 years, 10 months ago

This will be useful later!

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