*Thus the solution must not lose validity for any of the equations.*Select your options so that your calculations are simple and use any method that suits you.equations with suitable constants so that when the modified equations are added, one of the variables is eliminated.

This process is repeated until one variable and one equation remain (namely, the value of the variable).

From there, the obtained value is substituted into the equation with 2 variables, allowing a solution to be found for the second variable.

To solve for three unknown variables, we need at least three equations.

Consider this example: Being that the first equation has the simplest coefficients (1, -1, and 1, for appears in the other two equations: Reducing these two equations to their simplest forms: So far, our efforts have reduced the system from three variables in three equations to two variables in two equations.

However, we are always guaranteed to find the solution, if we work through the entire process.

The word "system" indicates that the equations are to be considered collectively, rather than individually.

Usually, though, graphing is not a very efficient way to determine the simultaneous solution set for two or more equations.

It is especially impractical for systems of three or more variables.

Now, we can apply the substitution technique again to the two equations of 2, 4, and 12, respectively, that satisfy all three equations.

While the substitution method may be the easiest to grasp on a conceptual level, there are other methods of solution available to us.

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