Solving Simultaneous equations

Question

Hi,

Consider following equations:
 a_1x+b_1y=c_1 
 a_2x+b_2y=c_2

To get the value of x & y
 x=(c_1b_2-c_2b_1)/(a_1b_2-a_2b_1) 
 y=(a_1c_2-a_2c_1)/(a_1b_2-a_2b_1) 
 This is Cramer's rule 
Remember to reduce the equations in the above mentioned form.

For example:
2x+3y=1
7x+6y=8

According to the Cramer's rule,
 x=(1*6-8*3)/(2*6-7*3)=2 
 y=(2*8-7*1)/(2*6-7*3)=-1

Regards,

nsspaz151 · Answer

Seems like a lot of work. Would it not be quicker in your example to solve as follows:

2x+3y=1 
 7x+6y=8

Multiply top equation by two:

4x+6y=2 
and
 6y=2-4x

Substitute in second equation, solve for x

7x+2-4x=8 
 3x=6 
 x=2

Solve for y in either

2*2+3y=1 
 4+3y=1 
 3y=-3 
 y=-1

cyberjadugar · Answer

Hi,

It is useful in long run. One has to practice to get used to this forumla, then it could be used conviniently.

Regards,

aemden1 · Answer

Hey cyberjadugar,

This is really handy! Thanks for sharing this. It seems familiar, but I've been out of school for so long, I would swear that most of things I am re-learning I was never taught! I'm going to check if it works with fractions. Kudos man!

Cheers,

toni

rswamina · Answer

Will we consider the signs of the coefficients while multiplying ?

ReedArnoldMPREP · Answer

For what it's worth, gang, I got a 760 on this thing (and have a minor in math) and have never seen this formula before. It seems easier to just... get good at substitution and/or combination.

IanStewart · Answer

Yeah, I completed grad school in math, and I doubt any mathematician I know would ever use Cramer's rule to solve GMAT-level 2-equations/2-unknowns problems. It's easy to see where Cramer's rule comes from, and then it's easy to see why it's actually a bad idea to use it for simple problems. If you have this two equation/two unknowns problem:

ax + by = p
gx + hy = r

then if we want to solve for y, we can just get the same 'coefficient' in front of x, and subtract one equation from the other. So we can multiply the first equation by g, and the second by a, to get agx in both:

agx + bgy = gp
agx + ahy = ar

and subtracting equations, the 'agx' term will vanish:

bgy - ahy = gp - ar
y(bg - ah) = gp - ar
y = (gp - ar)/(bg - ah)

and that's Cramer's rule. Notice though if you had these two equations:

154x + 29y = 53
77x + 14y = 25

then if you plug into Cramer, you're doing this calculation: (77*53 - 154*25)/(29*77 - 14*154), which is bananas, at least if you don't see the useful factorization of 77 throughout. If you avoid Cramer, and just multiply the second equation by 2 and subtract equations, you simply do:

154x + 29y = 53
154x + 28y = 50

y = 3

So when you use Cramer, you're doing exactly the same thing that you would do when you subtract equations,  if you subtract equations in the worst possible way . You're always doing at least as much work as you'd do subtracting equations, but we can often use least common multiples, or many other simple shortcuts, and if you commit to using Cramer, you're committing to never noticing any of those time-saving opportunities.

Solving Simultaneous equations

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