Solving Problems Using Simultaneous Equations

Solving Problems Using Simultaneous Equations-77
In our example equation set, for instance, we may add ) into one of the original equations.In this example, the technique of adding the equations together worked well to produce an equation with a single unknown variable.

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The step-by-step example shows how to group like terms and then add or subtract to remove one of the unknowns, to leave one unknown to be solved.

It involves what it says − substitution − using one of the equations to get an expression of the form ‘y = …’ or ‘x = …’ and substituting this into the other equation.

Let’s take our two-variable system used to demonstrate the substitution method: One of the most-used rules of algebra is that you may perform any arithmetic operation you wish to an equation so long as you do it equally to both sides.

With reference to addition, this means we may add any quantity we wish to both sides of an equation—so long as its the same quantity—without altering the truth of the equation.

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.

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 gives an equation with just one unknown, which can be solved in the usual way.This value is then substituted in one or other of the original equations, giving an equation with one unknown.It is especially impractical for systems of three or more variables.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.This is then substituted into one of the otiginal equations.The 2 lines represent the equations '4x - 6y = -4' and '2x 2y = 6'. Because the graphs of 4x - 6y = 12 and 2x 2y = 6 are straight lines, they are called linear equations.What about an example where things aren’t so simple?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.An option we have, then, is to add the corresponding sides of the equations together to form a new equation.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.


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