A System of Equations | Problem 355

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[@XJWill1]

I think the straightforward method is also among the easiest. Just solve the first equation for w, sub that into the second equation and get a quadratic in z which is easily solved. Then sub the solution for z back into the solution for w in terms of z that was obtained previously by solving the first equation for w.

[@tygrataps]

Interesting! Did you try the option of subtracting (1) and (2) and getting the difference of two squares?

[@aplusbi]

I haven't but that's great idea!

[@neuralwarp]

Can you do something on Quaternions?

[@trojanleo123]

Did using 4th method by subtraction and also division.

[@key_board_x]

(1): z² - zw = 1 + 7i (2): w² - zw = - 4 - 3i --------------------------------substraction (z² - zw) + (w² - zw) = (1 + 7i) + (- 4 - 3i) z² - zw + w² - zw = 1 + 7i - 4 - 3i z² - 2zw + w² = - 3 + 4i (z - w)² = - 3 + 4i ← memorize this result → let's find a complex number n, such as: n² = - 3 + 4i n = a + ib n² = a² + 2abi + i²b² n² = a² - b² + 2abi → to be compared to: - 3 + 4i 2abi = 4i ab = 2 b = 2/a a² - b² = - 3 a² - (2/a)² = - 3 a² - (4/a²) = - 3 (a⁴ - 4)/a² = - 3 a⁴ - 4 = - 3a² a⁴ + 3a² - 4 = 0 → let: A = a² → where: A > 0 A² + 3A - 4 = 0 Δ = (3)² - (4 * - 4) = 9 + 16 = 25 A = (- 3 ± 5)/2 → we keep only the positive value A = (- 3 + 5)/2 A = 1 a² = 1 a = ± 1 First case: a = 1 b = 2/a b = 2 → recall: n = x + iy n = 1 + 2i → n² = 1 + 4i - 4 = - 3 + 4i Second case: a = - 1 b = 2/a b = - 2 → recall: n = x + iy n = - 1 - 2i → n² = [- (1 + 2i)]² = (1 + 2i)² = - 3 + 4i Restart: (z - w)² = - 3 + 4i → recall: n = 1 + 2i → where: n² = - 3 + 4i (z - w)² = (1 + 2i)² z - w = ± (1 + 2i) First case: z - w = + (1 + 2i) z - w = 1 + 2i → recall (1) z² - zw = 1 + 7i z.(z - w) = 1 + 7i → see above z.(1 + 2i) = 1 + 7i z = (1 + 7i)/(1 + 2i) z = (1 + 7i).(1 - 2i)/[(1 + 2i).(1 - 2i)] z = (1 - 2i + 7i - 14i²)/[1 - 4i²] z = (1 + 5i + 14)/[1 + 4] z = (15 + 5i)/5 z = 3 + i → recall: z - w = 1 + 2i w = z - 1 - 2i w = 3 + i - 1 - 2i w = 2 - i Second case: z - w = - (1 + 2i) z - w = - 1 - 2i → recall (1) z² - zw = 1 + 7i z.(z - w) = 1 + 7i → see above z.(- 1 - 2i) = 1 + 7i z = - (1 + 7i)/(1 + 2i) z = - (1 + 7i).(1 - 2i)/[(1 + 2i).(1 - 2i)] z = - (1 - 2i + 7i - 14i²)/[1 - 4i²] z = - (1 + 5i + 14)/[1 + 4] z = - (15 + 5i)/5 z = - 3 - i → recall: z - w = - 1 - 2i w = z + 1 + 2i w = - 3 - i + 1 + 2i w = - 2 + i First solution: z = 3 + i w = 2 - i Second solution: z = - 3 - i w = - 2 + i

[@dorkmania]

In method 2, why not evaluate the fraction and get (w/z) = (1 - i)/2 or (z/w) = (1+ i)

[@aplusbi]

sure

[@mcwulf25]

Method 2 you should have -17 before you gave up!!! Method 4: divide (1) by (2) and divide top and bottom by zw. (z/w - 1)/(w/z - 1) = (1+7i)/(-4-3i) Put t = z/w and cross multiply (t-1)(-4-3i) = (1/t - 1)(1+7i) t(t-1)(-4-3i) =(1-t)(1+7i) If t 1 (this must be the case) t = (1+7i)/(-4-3i) Proceed as per your #2. Method 5: Add and subtract. You get (z-w)^2 = (-3+4i) (z+w)(z-w) = (5+10i) Now divide (z-w)/(z+w) = (-3+4i)/(5+10i) This gives z in terms of w. Substitute and solve.

[@aplusbi]

Nice. With #5 you could also square root the first eqn and then substitute into the 2nd. This should give you a linear system

[@mcwulf25]

@aplusbi This could be method 6 😁. Requires knowing how to square root a complex number- which I know you have demonstrated.

[@trojanleo123]

I'm okay with long videos. Is there a reason you feel constrained to keep your videos on the shorter side?

[@aplusbi]

A lot of people are turned off by long videos and avoid watching them (short attention span, being used to watch shorts/tiktok/insta stories...) I try to keep them under 10 minutes. That's probably a psychological limit.