How do you solve systems by elimination?

Answer 1

Select one equation from a system of linear equations.

Select a second equation.

Cross-multiply the equations by the coefficient of one of the variables and subtract one equation from the other. The resulting equation will have one fewer variable.

Select another "second" equation and repeat the process for the same variable until you have gone through all the remaining equations.

At the end of the process you will have one fewer equation in one fewer variable. That variable will have been eliminated from the system of equations.

Repeat the whole process again with another variable, and then another until you are left with one equation in one variable.

That, then, is the value of that variable.

Substitute this value in one of the equations from the previous stage to find the value of a last variable to be eliminated. Work backwards to the first variable.

Done!

Unless: when you are down to one equation it is in more than one variable. In this case your system of equations does not have a unique solution. If there are n variables in your last equation then n-1 are free to take any value. These do not have to be from those in the last equation.

or

when you are down to one variable you have more than one equation. If the equations are equivalent (eg 2x = 5 and -4x = -10), you are OK. Otherwise your system of equations has no solution.