jaybrium wrote:BTG Community,
When solving a system of equations, the rule of thumb is "if you have n variables in the system, then you need n distinct equations."
Does this really mean that you need "n distinct first degree equations?" Say you have 3 variables, and 3 distinct equations; if one, two, or three of these equations have the variables raised to different powers, can you still solve for the three variables?
Thanks as always!
A lot of GMAT prep materials mention this 'rule' as though it were just that- a rule. It's not. We can, if we introduce some severe restrictions, come up with a rule that is guaranteed to be correct, but especially at the hard level of the GMAT, you'll see many exceptions to the 'n unknowns, n equations rule'.
First I'll point out some of the exceptions:
Find c:
a+b+c = 7
a+b = 5
Here we only have 2 equations, 3 unknowns, but it's still easy to find c=2. No chance of finding a or b though.
Find x:
y= x^2
y = x + 2
Here we have two equations, two unkowns, but there are two different solutions for x and y (this is easier to see if you know co-ordinate geometry; the first equation is a parabola, the second a line which intersects the parabola in two points).
Find x:
x + y = 5
y + z = 7
x + 2y + z = 12
Here we have three equations, three unknowns, but an infinite number of different solutions for x, y and z. You might notice that the third equation is just the sum of the first two; we can derive the third from the first two equations, so it really isn't any new information at all. Fundamentally, the above is really just two equations, three unknowns.
Find x/y:
3x = 4y
Here we only have one equation, two unknowns. There's no way to find x, or to find y here, but we still might be able to solve for some combination of x and y. The above is just a standard ratio equation, and it's easy to find x/y by dividing.
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The real "rule" in mathematics is this: If you have n *independent* *linear* equations and n unknowns, you can solve for all n of your unknowns.
* By "linear", we mean "first degree" - you have no exponents, you aren't multiplying or dividing any variables with other variables, etc. With only two unknowns, a linear equation is one that can be rewritten in the form y = mx + b (hence the name 'linear'). If you have non-linear equations (like y = x^2, or xy = 1), then there is no general rule that you can use.
* By "independent", we mean that you can't derive any one equation from the others. These equations are not independent, for example:
y = 2x - 3
3y = 6x - 9
because you can derive the second by multiplying the first by 3. With only two equations, two unknowns, 'independent' essentially means that one equation is not an exact multiple of the other. With more unknowns, it can be non-trivial to check that your equations are independent; unless you use techniques from linear algebra, you might need to start solving.
So the short answer to your question is yes- the equations need to be first degree equations for there to be any chance of having a rule. But still, even with first degree equations, things aren't always straightforward, and the GMAT often tests the exceptions to the simplistic guidelines some books teach about algebra.
