non_linear_systems

non_linear_systems - Non-linear Systems of Equations in two...

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Non-linear Systems of Equations in two variables Example 1 : See [9.4, #7] Question : Solve the following system of equations and interpret the solution graphically. { = y + x 1 = + x 2 y 2 25 . Solution : Substituting = y + x 1 from the first equation into the second equation gives: = + x 2 ( ) + x 1 2 25. This equation is equivalent to = + x 2 ( ) + + x 2 2 x 1 25, that is, = + - 2 x 2 2 x 24 0, which, in turn, is equivalent to = + - x 2 x 12 0, and to = ( ) + x 4 ( ) - x 3 0. Thus we obtain the two solutions = x - 4 and = x 3. The equation = y + x 1 can be used to obtain the corresponding y values which are = y - 3 and = y 4 respectively. Thus we obtain two solutions for the system of equations: = x - 4, = y - 3 and = x 3, = y 4. _____________ Graphical illustration of the solutions : The graph of the equation = + x 2 y 2 25 is a circle with its centre at the origin and with aradius of 5 units. This circle is drawn in red in the following picture. The line with equation = y + x 1 is drawn in purple . The solutions of the system of equations correspond to the coordinates of the two points of intersection of these two graphs. Example 2 : [9.4, #18] Question : Solve the following system of equations and interpret the solution graphically. { = y x 2 = y - 3 x 2 . Solution : The variable y can be eliminated by equating the right-hand sides of the two equations to give:
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= x 2 - 3 x 2 ------- (i). This equation is equivalent to = - + x 2 3 x 2 0. The left-hand side of this quadratic equation can be factored to give: = ( ) - x 1 ( ) - x 2 0. Hence equation (i) has the two solutions: = x 1, = x 2. The corresponding y values obtained by using the equation = y x 2 are: = y 1, = y 4 respectively. There are two solutions for the system of equations: = x 1, = y 1 and = x 2, = y 4. ______________ Graphical illustration of the solutions : In the following picture the parabola = y x 2 is drawn in red , while the line = y - 3 x 2 is drawn in purple . The solutions correspond to the coordinates of the
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non_linear_systems - Non-linear Systems of Equations in two...

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