# Simultaneous Equations Problem Solving

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.

In this example, the technique of adding the equations together worked well to produce an equation with a single unknown variable.

What about an example where things aren’t so simple?

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.

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

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The word "system" indicates that the equations are to be considered collectively, rather than individually.Since each equation is an expression of equality (the same quantity on either side of the sign), adding the left-hand side of one equation to the left-hand side of the other equation is valid so long as we add the two equations’ right-hand sides together as well.In our example equation set, for instance, we may add ) into one of the original equations.Perhaps the easiest to comprehend is the substitution method.Take, for instance, our two-variable example problem: In the substitution method, we manipulate one of the equations such that one variable is defined in terms of the other: Then, we take this new definition of one variable and substitute it for the same variable in the other 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.Consider the following equation set: We could add these two equations together—this being a completely valid algebraic operation—but it would not profit us in the goal of obtaining values for : The resulting equation still contains two unknown variables, just like the original equations do, and so we’re no further along in obtaining a solution.However, what if we could manipulate one of the equations so as to have a negative term that would cancel the respective term in the other equation when added?The process is repeated until the values of all When we have more variables to work with, we just have to remember to stick to a particular method, and keep on reducing the number of equations or variables.We will solve the following system of equations using both approaches: Elementary row operation or Gaussian elimination is a popular method for solving system of linear equations.

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