Reducing these two equations to their simplest forms: Now, we can substitute this definition of x where x appears in the other two equations:
Consider this example:īeing that the first equation has the simplest coefficients (1, -1, and 1, for x, y, and z, respectively), it seems logical to use it to develop a definition of one variable in terms of the other two. To solve for three unknown variables, we need at least three equations. 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. Or, to save us some work, we can plug this value (6) into the equation we just generated to define y in terms of x, being that it is already in a form to solve for y:Īpplying the substitution method to systems of three or more variables involves a similar pattern, only with more work involved.
Now that x is known, we can plug this value into any of the original equations and obtain a value for y.
Now that we have an equation with just a single variable ( x), we can solve it using “normal” algebraic techniques: In this case, we take the definition of y, which is 24 - x and substitute this for the y term found in the other equation: Then, we take this new definition of one variable and substitute it for the same variable in the other equation. In the substitution method, we manipulate one of the equations such that one variable is defined in terms of the other: Take, for instance, our two-variable example problem: Perhaps the easiest to comprehend is the substitution method. Several algebraic techniques exist to solve simultaneous equations. Solving Simultaneous Equations Using The Substitution Method In a three-variable system, for example, the solution would be found by the point intersection of three planes in a three-dimensional coordinate space-not an easy scenario to visualize. It is especially impractical for systems of three or more variables. Usually, though, graphing is not a very efficient way to determine the simultaneous solution set for two or more equations. There is only one point where the two linear functions x + y = 24 and 2x - y = -6 intersect (where one of their many independent solutions happen to work for both equations), and that is where x is equal to a value of 6 and y is equal to a value of 18. Plotted on a graph, this condition becomes obvious:Įach line is actually a continuum of points representing possible x and y solution pairs for each equation.Įach equation, separately, has an infinite number of ordered pair ( x, y) solutions. Example:įor this set of equations, there is but a single combination of values for x and y that will satisfy both.Įither equation, considered separately, has an infinitude of valid (x,y) solutions, but together there is only one. 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. What are Simultaneous Equations and Systems of Equations?
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