A Very tricky Harvard University Entrance Interview Exam | Find xy=?

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[SanePerson]

Add the two equations to get (x + y)² = 49 ⇒ x + y = ±7 or y = ±7 – x, two straight lines, both with slopes of –1 and x-axis intercepts at ±7. Subtract the second equation from the first to get x² – y² = 21, a hyperbola that intersects the x axis at x = ±√21. The intersection of the two lines with the hyperbola occur at (5, 2) and (–5, –2) which, to be honest, one can probably guess in less that a minute by inspection of the two initial equations. It's takes a little bit of time and effort to PROVE there are no other intersections, but it's graphically evident and not all that difficult otherwise.

[xualain]

Add the two equations
x^2+2xy+y^2=49 —>x+y=7 or x+y=-7
If x+y=7 from eq(1) x^2+xy=x*(x+y)=7x=35 hence x=5
from eq(2) yx+y^2=y*(x+y)=7y=14 hence y=2
Similarly for x+y=-7 from eq(1) -7*x=35 hence x=-5
From eq(2) -7*y=14 hence y=-2

[moemirza]

Subtract Eq. 2 from 1 and
(x - y) (x + y) = 3 * 7 & -3 * -7
x - y = 3 & -3
x + y = 7 & -7
Add & x = 5 . -5 y = 2, -2
other combos like 1 * 21 and -1 * -21 does not satisfy.

[key_board_x]

x² + xy = 35
y² + xy = 14

(x² + xy) + (y² + xy) = 35 + 14
x² + xy + y² + xy = 49
x² + 2xy + y² = 49
(x + y)² = 49
x + y = ± 7

x² + xy = 35
y² + xy = 14

(x² + xy) - (y² + xy) = 35 - 14
x² + xy - y² - xy = 21
x² - y² = 21
(x + y).(x - y) = 21
x - y = 21/(x + y) → recall: x + y = ± 7
x - y = 21/± 7
x - y = ± 3

(x - y)² = x² - 2xy + y² → recall the previous result:
(± 3)² = x² - 2xy + y²
9 = x² - 2xy + y²
x² - 2xy + y² = 9 → recall the memorized result: x² + 2xy + y² = 49
x² + 2xy + y² = 49

(x² - 2xy + y²) - (x² + 2xy + y²) = 9 - 49
x² - 2xy + y² - x² - 2xy - y² = - 40
- 4xy = - 40
→ xy = 10

[НеллиПшено]

Решаем методом устного счета.
35=25+10
14=4+10
25=5^2
x=5
4=2^2
у=2
Good luck!

[walterwen]

Harvard University Entrance Interview Exam: x² + xy = 35, y² + xy = 14; x, y =?
35 > 14, x² + xy = 35 > y² + xy = 14; x² > y²
(x² + xy) + (y² + xy) = 35 + 14, x² + 2xy + y² = (x + y)² = 49, x + y = ± 7
(x² + xy) – (y² + xy) = 35 – 14, x² – y² = (x – y)(x + y) = 21, x – y = 21/(x + y) = ± 3
x + y = 7, x – y = 3: 2x = 10, x = 5, y = x – 3 = 5 – 3 = 2; x² = 25 > y² = 4
x + y = 7, x – y = – 3: 2x = 4, x = 2, y = x + 3 = 5; x² = 4 < y² = 25, Rejected
x + y = – 7, x – y = 3: 2x = – 4, x = – 2, y = x – 3 = – 5; x² = 4 < y² = 25, Rejected
x + y = – 7, x – y = – 3: 2x = – 10, x = – 5, y = x + 3 = – 2; x² = 25 > y² = 4
x = 5, y = 2 or x = – 5, y = – 2
Answer check:
x = ± 5, y = ± 2
x² + xy = (± 5)² + (± 5)(± 2) = 25 + 10 = 35; Confirmed
y² + xy = (± 2)² + (± 2)(± 5) = 4 + 10 = 14; Confirmed
Final answer:
x = 5, y = 2 or x = – 5, y = – 2

[jlp]

A usual; your solution is overcomplicated : add the equations : (x + y)² = 49 so x+y = (+-)7. Then substract the equations : x² - y² = (x-y)(x+y) = 21. We already know x+y, now we get x-y, so x and y. A video for this solution should last 2 minutes.