Algebra2Ch1cont.part15

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The next section is named 'Systems of Two Linear Equations' "A 'system of equations' involves more than one 'equation' with more than one 'variable'. In this section, we will look at systems of two linear equations in two variables. A 'solution' to a 'system of two equations' in 'two equations in two variables' is an ordered pair that satisfies 'both' equations. Typically, there will be only one such solution although other possibilities will be discussed later. The solution can be found either 'graphically or algebraically'." 'Algebraic Solutions' "There are two basic 'algebraic methods' for solving a system of two linear equations, the 'elimination method' and the 'substitution method'. The method you use is largely a matter of personal preference. However, sometimes the form of the given equations makes one method easier than the other." "Elimination Method' "For two equations in the form {Ax +BY =C and DX +Ey = F, the elimination method may be easiest. The idea of the method is to eliminate one of the variables by adding the two equations. The resulting equation can be solved for the remaining variable. This value can then be substituted into either equation to find the value of the second variable". Example 1.25 Solve the system: {5x =2y = 9 and x -2y = 3 Solution In this case, when the two equations are added, the y-terms will "drop out" of the equation because they have opposite coefficients. 5x +2y = 9 substituting the 2 5(2) + 2y = 9 + x -2y = 3 10 + 2y = 9 ____________ 2y = -1 6x = 12 y = -0.5 x =2 The solution is (2, - 0.5) "This was an easy example; the coefficients of the y-terms in the two given equations were opposites. When this does not happen, you can multiply one or both of the equations by appropriate constants so that the coefficients of one variable are opposites. Which variable you eliminate does not matter. Sometimes an easy choice will be obvious. If not, you can multiply the first equation by the coefficient of x in the second equation and multiply the second equation by the 'opposite' of the coefficient of x in the first equation. This will eliminate x. (Go to next lesson)