10. Simultaneous Equations
A set of N independent equations in N unknowns is called a system of simultaneous equations.
This system can then be solved to determine the values of the N unknown variables.
To solve a system of simultaneous equations, simply eliminate one variable at a time till you are left with a simple linear equation with one variable.
Then solve that linear equation and substitute back to solve for the other variables one at a time.
e.g. $2x+y=3$, $x+2y=3$
First, we notice that the equations are independent. You cannot get one equation by multiplying the other with a constant.
Next, we multiply the second equation by 2. Now we get, $2x+y=3$, and $2x+4y=6$
Subtracting the first equation from the second, we get, $(2x-2x)+(4y-y)=(6-3)$ or $3y=3$
Solving this, we get $y=1$, substituting in the first equation, we get $2x+1=3$, which gives $x=1$
So we have solved the system of equations, giving $x=1$, and $y=1$.
To reiterate, in general, you need N independent equations to be able to solve for N unknowns.
However, in the GMAT, you will never be asked to solve a system of more than three equations at the most.
Some trick questions might require you to perform a transformation before you can solve them.
If you set $x=1/a$ and $y=1/b$, you end up with the same set of equations as from the previous example.
We already know the solution there was $x=y=1$. We now substitute back (we want the values for a and b, not x and y)
So we get, $a=1/x=1/1=1$ and $b=1/y=1/1=1$