# Example 1 consider the system defined by the

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Example 1: Consider the system defined by the following 2 equations: (i) 2 x + y = 30 and (ii) y = 4 x. We can substitute 4 x for y into the first equation to get 2 x + (4 x ) = 30, and so x = 30/6 = 5, and y = 4(5) = 20. Since the 2 equations are independent, they give a unique solution: no other pair of values than (5, 20) will satisfy both (i) and (ii). M3-1 M ATH M ODULE S olving Linear Equation Systems 3

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M3-2 MATH MODULE 3: SOLVING LINEAR EQUATION SYSTEMS Example 2: Consider the system defined by the following 2 equations: (i) 4 x + 2 y = 60 and (ii) y = 30 – 2 x. In this case, if we substitute 30 – 2 x for y into the first equation, we end up with 0 = 0. This is because the 2 equations are dependent : if we double every term in equation (ii) and rearrange, we end up with equation (i). Hence the graphs of the 2 equations exactly coincide , and the number of solutions is infinite : any ( x , y ) pair that satisfies (i) will satisfy (ii). Example 3: Consider the system defined by the following 2 equations: (i) y = 30 – 2 x, and (ii) y = 20 – 2x. In this case, where the 2 equations are inconsistent, their graphs are parallel and therefore never intersect. Hence there is no ( x , y ) pair that satisfies both (i) and (ii). You may wish to graph these three examples, for additional practice and to ensure that you have a clear visual image of what independent, dependent, and inconsistent systems of equations look like. 2. Exercises 1. For the following sets of equations, state whether they are independent, dependent or inconsistent, and give the solutions for x and y if the equations are independent. (a) (i) 3 y = 2 x – 6 , (ii) 4 x = 12 + 6 y (b) (i) 10 = 3 x + 2 y, (ii) 20 = 3 x + 4 y (c) (i) y = 40 – x, (ii) y = x (d) (i) y = 40 – x, (ii) 3 y = x (e) (i) y = 40 – x, (ii) x = (1/3) y (f) (i) 3 y = 90 – 2 x, (ii) x = 30 – 1.5 y (g) (i) y = 30 – 2 x, (ii) y = 15 – 0.5 x (h) (i) y = 60 – 2 x, (ii) y = 30 – 0.5 x 2. Solve the following pairs of equations, and compare your results for (a) with (b), (c), and (d):
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