How to Solve a Non-Linear System of Equations
Linear systems of equations contain only x and y terms, while non-linear systems contain more complicated terms such as exponential functions and polynomials. Non-linear systems model many processes in nature, particularly growth and decay. In geometry, you must solve a system of non-linear equations to find the intersection points of conic sections. You find the solutions of non-linear systems using the substitution method or a combination of substitution and graphing. Systems of non-linear equations sometimes have more then one solution.
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Instructions
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Choose the equation to easily isolate one of the variables. For instance, in the systems xy + y = 22 and xy + (xy)^2 = 420, you can quickly isolate x in the first equation.
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Isolate one of the variables in the chosen equation. For example, when you use algebra to isolate x in the equation xy + y = 22, you obtain x = (22-y)/y.
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Plug the expression into the other equation to obtain a new equation that contains only one variable. For example, replace x with (22-y)/y in the equation xy + (xy)^2 = 420. This gives you (22 - y) + (22 - y)^2 = 420.
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Use algebra to simplify the resulting equation so that you can more easily solve it. For instance, when you expand and combine like terms in the equation (22 - y) + (22 - y)^2 = 420, you arrive at y^2 - 45y + 86 = 0
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Solve the equation using either algebra or a computerized solver, such as a graphing calculator. For example, solve the equation y^2 - 45y + 86 = 0 using the quadratic formula by hand. The solutions are y = 2 and y = 43.
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Plug the solutions into the simpler of the two original equations to solve for the missing variable. For example you find x = 10 when you plug y = 2 into xy + y = 22. You also find x = -21/43 when you plug y = 43 into xy + y = 22. The two solutions in this example are x = 10 and y = 2, as well as x = -21/43 and y = 43.
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References
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