# Simultaneous Equations Problem Solving

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equations with suitable constants so that when the modified equations are added, one of the variables is eliminated.

Once this is done, the system will have effectively been reduced by one variable and one equation.

As with substitution, you must use this technique to reduce the three-equation system of three variables down to two equations with two variables, then apply it again to obtain a single equation with one unknown variable.

The solution to a linear system is an assignment of numbers to the variables that satisfy every equation in the system.

The terms simultaneous equations and systems of equations refer to conditions where two or more unknown variables are related to each other through an equal number of equations.

For this set of equations, there is but a single combination of values for is equal to a value of 18.In this case, we take the definition of : Applying the substitution method to systems of three or more variables involves a similar pattern, only with more work involved.This is generally true for any method of solution: the number of steps required for obtaining solutions increases rapidly with each additional variable in the system.Then, the system would reduce to a single equation with a single unknown variable just as with the last (fortuitous) example.If we could only turn the is easily determined: Using this solution technique on a three-variable system is a bit more complex.For a given system, we could have one solution, no solutions or infinitely many solutions.Using the process of substitution​ may not be the quickest nor the easiest approach for a given system of linear equations.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.