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Solving System of Equations(Substitution Method)

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In mathematics, a system of linear equations (or linear system) is a collection of linear equations involving the same set of variables. For example,

is a system of three equations in the three variables x, y, z. A solution to a linear system is an assignment of numbers to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by

since it makes all three equations valid.




Substitution method


Systems of simultaneous equations can be hard to solve unless a systematic approach is used. A common technique is the substitution method: Find an equation that can be written with a single variable as the subject, in which the left-hand side variable does not occur in the right-hand side expression. Next, substitute that expression where that variable appears in the other equations, thereby obtaining a smaller system with fewer variables. After that smaller system has been solved (whether by further application of the substitution method or by other methods), substitute the solutions found for the variables in the above right-hand side expression. (Our solved example in mathguru.com uses this concept).


In this set of equations

x is made the subject of the second equation:

then, this result is substituted into the first equation:

After simplification, this yields the solutions

and by substituting this in x = −2y the corresponding x values are obtained. The two solutions of the system of equations are then:





The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.