用绘制图像的方法求解非线性方程

There are different methods to solving systems of equations. You've already learned about graphing systems of linear equations, but now we're going to dive into graphing systems of nonlinear equations.

This is just one method of solving systems of nonlinear equations! You can also use elimination and substitution.

Before we begin, let's establish a definition for what a non-linear system of equations is.

A non-linear system of equations is a system of two or more equations, where at least one equation is non-linear.

Non-linear equations are equations where the power is not equal to $1$, such as quadratic functions ($x^2$), cubic functions ($x^3$), and square root functions ($x^{\frac12}$).

Graphing Systems of Nonlinear Equations

The rules for graphing systems of linear and nonlinear equations are very similar. The point of intersection for two graphed equations is the solution to the system of equations. This definition is the same as we saw with linear systems of equations. The main difference, however, is there can be multiple points of intersection with non-linear systems of equations!

Imagine a quadratic function (U-shaped) and a linear function (a line). If the two functions were to meet (at a point that's not the vertex), the linear function would intersect the quadratic function twice.

To review, check out our lesson on how to graph functions.

Look at the following graph:

The blue function represents $x+1$. The red function represents $x^2+2x-2$.

As we can see from the graph, these two functions intersect at two points, $(-2.3, -1.3)$ and $(1.3, 2.3)$. These two points of intersection represent the solution to the system of equations.

So, the system of non-linear equations,

$$y =\left\{\begin{matrix} x+1\,\,\,\,\\ -4x-2\,\,\,\, \end{matrix}\right.$$

...has a solution of $(-2.3, -1.3)$ and $(1.3, 2.3)$.

When we are evaluation a system of non-linear equations, we may not always have two solutions. In the previous example, we had two solutions because the linear equation crossed the quadratic equation.

In this next example, let's consider what happens when we have a system of non-linear equations, a linear function and a cubic function.

The blue linear function is $x+1$ and the red cubic function is $x^3+1$. The two functions meet at three points, $(-1,0)$, $(0,1)$, and $(1,2)$. We know that the solution to a non-linear system of equations is the points where the two functions meet.

That means that $(-1,0)$, $(0,1)$, and $(1,2)$ are the solutions to the following system of equations:

$$y =\left\{\begin{matrix} x+1\,\,\,\,\\ x^3+1\,\,\,\, \end{matrix}\right.$$

There are times when a system of non-linear equations may not have a solution! This would mean that the two graphed equations do not meet at all. Here is an example:

Try an example on your own!

Example

Solution

If we were to graph the two equations, we would see the following graph:

We know that the solution to a system of non-linear equations is when the two functions intersect. In this case, the two points of intersection are: $(-0.5, -1)$ and $(1,2)$.

(Video) Solving a System of Non-Linear Equations

by Tara Jones Math Videos

This video by Tara Jones Math Videos walks you through an example of using graphing to solve a system of nonlinear equations.

Summary

To solve a system of nonlinear equations by graphing, we want to find the intersections of the curves. The point(s) of intersection make up the solutions.

Solve the following system of equations:

$$x^2-y=2$$ $$2x-y=-1$$

The first equation has a degree of $2$, so it's a quadratic function, shaped like a parabola. The second equation has a degree of $1$, so it's a linear function. So, we are looking for the intersection of a parabola and a line.

Let's first manipulate the first equation to make it into standard form.

$$\begin{align} x^2-y&=2\\ -y &= -x^2 +2\\ y &= x^2 -2\\ \end{align}$$

Let's graph this parabola.

Now, we're going to tackle the linear equation. Let's convert it to slope-intercept form.

$$\begin{align} 2x-y&=-1\\ -y &= -2x -1\\ y &= 2x +1\\ \end{align}$$

We can graph this linear equation on the same graph as the parabola.

These two equations intersect at two points: $(-1,-1)$ and $(3,7)$.

Graphing Systems of Nonlinear Equations

If we were to graph two equations,

$$\begin{align} y &= x^2 -3 \\ y&= 4x -3 \\ \end{align}$$

we could find the points of intersection. The points where the two graphs meet are the solutions to the system.

We can see that the graphs meet at $(4,13)$ and at $(0,-3)$. Those are the points of intersection.