How to Solve Simultaneous Nonlinear Equations
In layman's term, a "linear" equation is one which appears as a straight line when you graph it. Expectedly, graphing nonlinear equations results in arced, curved segments. If you have a math problem which involves "simultaneous" nonlinear equations--also known as a "system" of nonlinear equations--it means that you have two nonlinear graphs which intersect with one another at at least one point. Calculating where precisely the graphs intersect is relatively straightforward, although it does require some concentration.
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Instructions
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Set your nonlinear equations equal to one another and simplify to combine like terms. Combine the equations "y = x^2 - x + 2 " and "y = 2x^2 - 6x +6," as follows: x^2 - x + 2 = 2x^2 - 6x + 6. Subtract "x^2" from both sides: x^2 - x^2 - x + 2 = 2x^2 - x^2 - 6x +6, or - x + 2 = x^2 - 6x + 6. Add "x" to both sides: - x + x + 2 = x^2 - 6x + x + 6, or 2 = x^2 - 5x + 6. Subtract "two" from both sides as follows: 2 -2 = x^2 - 5x + 6 - 2, or 0 = x^2 - 5x + 4.
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Solve your combined equation by factoring. You can factor any equation x^2 + (a+b)x + (ab) as (x + a)(x + b), keeping in mind that "a" and/or "b" may be negative. For your combined equation x^2 - 5x + 4, think of numbers ("a" and "b"), which you can multiple together to get positive four, but add together to get negative five--the only combination is negative four and negative one. As such, your factored equation is "(x - 4)(x - 1)," which means that whenever "x" is "four" or "one," your combined equation will equal zero--in other words, these are the horizontal coordinates where your original simultaneous equations intersect.
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Determine complete coordinates for your intersections by plugging your "x" values into either equation. For "y = x^2 - x + 2," do this as follows: y = (1)^2 - 1 + 2 = 1 - 1 + 2 = 2 and y = (4)^2 - 4 + 2 = 16 - 4 + 2 = 14. Check your answers by plugging the values into the second equation--for the example, 2x^2 - 6x + 6, like so: y = 2(1)^2 - 6(1) + 6 = 2 and y = 2(4)^2 - 6(4) + 6 = 2(16) - 24 + 6 = 14.
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