5. Graphical Solution of non-Linear Systems

A non-linear graph is a curve. This section assumes you already know the formulas for straight lines, circles, parabolas, ellipses and hyperbolas. You can refresh your memory in the Plane Analytic Geometry chapter.

In this section, we see how to solve non-linear systems of equations (those involving curved lines), using a graph. Our answers (as `x`-`y` coordinates) will be approximate, and we can improve our answer by using a graphics calculator or a computer package.

Example

Solve the system of equations graphically:

3xy = 4

y = 6 − 2x2

Answer

The graph of the parabola and straight line are as follows:

Graphs of `y = 3x-4` and `y=6-2x^2`.

We can see from the graph that there are 2 solutions, since there are 2 places where the graphs intersect. They are:

(a) At left, approximately (−3.2, −13).

Using computer graphing software (like Scientific Notebook), we can zoom in and find the solution correct to as many decimal places as we like. A few zooms gives us (-3.1085, -13.3255).

(b) At right, approximately (1.5, 1). Using a computer (or graphics calculator), we can zoom in on this intersection to get a better estimate of (1.6085, 0.8255).

Exercise 1

Solve graphically. Estimate your answer.

y = x2

xy = 4

Answer

y = x2 is a parabola.

xy = 4 is a hyperbola.

We see that the intersection point (the graphical solution) is at approximately (1.5, 2.5).

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Note: You can use the grapher on this page to get a better idea of what the graphs look like. You can also zoom in on the intersection points.

Exercise 2

Solve graphically. Estimate your answer:

y = 4xx2

y = 2 cos x

Answer

Graphs of `y = 4x-x^2` and `y=2cosx`: Intersection cosine curve and parabola

We see that the solutions are approximately
`(0.5, 1.8)` & `(4.2, −0.9)`.