How to solve equations fast ?

Question

Hi
Sometimes i get easy problems , i do thinking , planning right , and have very clear approach to it .Then I face an equation ( especially quadratic ) that really take a lot of time to solve .
 
I am wondering if there is some tips on how to solve equations fast ?

This is an example : it's not a GMAT question , but i think there is a similer one in OG .

a rectangle has x ,y dimensions and area of 60, 
 we draw a square has x side
another square has y side 
if the area of the two squares (combined) is 169
find x ,y

Please share your fastest approach to solve the equation in this example .

u1983 · Answer

Given :  xy=60 ,  x^2+y^2=169

(x+y)^2= 169+2*60 =289 ==>  x+y=17 ----(1)
 (x-y)^2= 169-2*60 =49 ==>  x-y=7 --------(2)
Hence solving for (1) & (2),  x=12  &  y=5 ............. following basics

foryearss · Answer

That's a fantastic approach i have not thought of, thanks !

SSM700plus · Answer

Hello My dear Friend,

It is not very difficult task. I think our approach has to be of Process of Elimination. This is where we would save more time.

So If I was given this question, I would have solved in the following ways:

You might very well know that area of rectangle is length * breadth = x*y=60 (given)

Find different combination of numbers such that when we multiply it will be equal to 60.
x be the first value & y be the second value.
a) 1* 60
b) 2*30
c) 3*20
d) 4*15
e) 5*12
f) 6 *10

If you are some what better in memorizing squares of number. And if you have solved some geometry problems.

You will easily find out  169 = 12^2 + 5^2 = 144 + 25 .

For more clarity on how to approach the problem using Process of elimination please post OG question in the respective forum.

Hope this helps.

ccooley · Answer

Lots of practice and drills. The MPrep Foundations of Math strategy guide is a good start for drills. Even if you already know the material, there are drills at the end of each chapter that will help you practice solving equations quickly. Khan Academy is another good (and in this case free) resource.

JaeTPU · Answer

For your specific example, 
x^2+y^2 = 169, xy=60 as given

Common quadratics:

(x+y)^2 = x^2 + y^2 + 2xy = 169 + 2(60) = 289
(x-y)^2 = x^2 + y^2 - 2xy = 169 - 2(60) = 49

Square root both,

x+y = 17
x-y = 7

Therefore, (x+y) + (x-y) = 2x = 17- 7 = 10
x =5

For questions like this it is best to get yourself familiar with the fact that when you have x^2+y^2 and xy you can solve for (x+y)^2 and (x-y)^2.

HOWEVER, if you didn’t know this approach, you can make an educated guess based on the numbers. You have 169, and if you are familiar with perfect squares (I highly recommend you memorize perfect squares 1-20), you should recognize that 13*13=169, which is 12^2 + 5^2, which happens to also satisfy xy=60.

bumpbot · Answer

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How to solve equations fast ?

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How to solve equations fast ?

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