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.

e.g.$2/a+1/b=3$, $1/a+2/b=3$

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 $b=1/y=1/1=1$

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.

e.g.$2/a+1/b=3$, $1/a+2/b=3$

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 $b=1/y=1/1=1$

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