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Created: 21 Aug, 2010; Last Modified: 29 Apr, 2018 Algebra II - 01Simultaneous Linear EquationsAn equation which has linear polynomials of two variables is a linear equation in two variables or a simultaneous linear equation. Some examples are:
A simultaneous linear equation has the form: .......(1)
Each problem involving simultaneous linear equations contains a set of two equations which relate the two unknown quantities or variables. These equations have to be considered together to find the solutions. Solving simultaneous equationsTwo common methods for solving simultaneous equations are the method of elimination by substitution and the method of elimination by equating coefficients. Method of elimination by substitutionThe steps are:
Solve
Consider the 1st equation. Solving for x we get: Substituting this value of x in the second equation gives: Thus, y Method of elimination by equating coefficientsThe steps are:
Solve
Consider eliminating x from the equations. On inspection, if we multiply the 1st equation with Thus, we have Subtracting (2) from (1) gives; With y A shop sells two kinds of cycles – bicycles and tricycles. There are Let b be the number of bicycles, and t be the number of tricycles in the shop. Since there are Also, if we consider the total of the wheels being (1) and (2) form a pair of simultaneous equations. From (1) we get Substituting the value of b above into (2), we have, Substituting the value t Solve
In this case, it will be easier to solve the equations if we let
Thus, the original equations can be written in the form: Using the method of "elimination by equating coefficients" to first eliminate a, it can be seen that the targeted coefficient for a is Subtracting (2) from (1) gives: Thus, with b This, in turn, implies that
Of course, these kinds of problems can also be solved in the normal way without involving a and b, in which
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Otherwise, send an email to feedback@mentorials.com with subject line: "Feedback: Simultaneous Linear Equations". List of ReferencesBansal, RK, Concise Mathematics I.C.S.E., Part I – Class IX, New Delhi: Selina Publishers, 2005. FHSST, Mathematics Grades 10 - 12, ver 0, viewed 10 March, 2009, <http://www.fhsst.org>, 2008. Schultze, A, Elements of Algebra, NY, USA: The Macmillan Company, 1910. BibliographyBansal, RK, Concise Mathematics I.C.S.E., Part I – Class IX, New Delhi: Selina Publishers, 2005. Gupta, SD & Banerjee, A, ICSE Mathematics for Class 9, Patna, India: Bharati Bhawan, 2003. Schultze, A, Elements of Algebra, NY, USA: The Macmillan Company, 1910.
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