Nonlinear System of Equations

Nonlinear system of equations consists of more than one algebraic equations that must be solved together, whether linear or non-linear. In effect, we see each equation as a graph, and when we solve the equations, we are looking for the points where those graphs cross or touch.

For example:

You're looking at y = x2 – 2 and y = -x2 + 3. The intersections are marked by the circles. Below are a few more examples.

This is an intersection of y = x3 and y = x, at 3 points.

There are no intersections between y = x2 - 2 and y = 4x2 - 1.

In a special case, where two graphs touch at ALL points, then the two graphs are identical.

As you can see, graphs provide a clear an easy way to spot where the intersections are, but it is usually unable to provide an exact answer unless the line falls on a corner of a grid box when plot using a computer. Plotted by hand, it can only give you a good estimate but not the exact answer. To get the exact answer, you'll need to use algebraic methods to solve the system of equations.

Example 1
Give y = 2x2 + 3x and y = 9x2 – 2, solve for x and y up to 3 decimal places.

Example 1 Solution
First we list the equations out.

y = 2x2 + 3x (Equation 1)
y = 9x2 – 2 (Equation 2)

Then you need to substitute one equation into the other. In this case, I choose to substitute Equation 2 into Equation 1. If you choose to substitute Equation 1 to Equation 2, it's okay too.

Since we have a quadratic equation, we can use the quadratic formula to solve for x.

We find that there are 2 values for x. Now we can substitute these values into either one of the equations to determine the respective values of y. For convenience, I choose to use Equation 2. For the first value of x up to 3 decimal places,

So we get the coordinates (0.790, 3.617). Substituting the second value of x into Equation 2 up to 3 decimal places,

So we get the coordinates (-0.362, -0.823). To check our results, we can plug in the x values into Equation 1 to see if the resulting y values match. Alternatively, if we plot the equations and mark the points where they cross, we can roughly determine that we have obtained the correct answer.

Hence, the coordinates (0.790, 3.617) and (-0.362, -0.823) are the intersections of these equations.

Continue to Example 2

Continue to Example 3

Continue to Examples 4 to 6

Continue to Example 7

Continue to Examples 8 and 9

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