Halley’s Comet orbits the sun about once every 75 years. Its path can be considered to be a very elongated ellipse. Other comets follow similar paths in space. These orbital paths can be studied using systems of equations. These systems, however, are different from the ones we considered in the previous section because the equations are not linear.

Figure 1. Halley’s Comet (credit: “NASA Blueshift”/Flickr)

In this section, we will consider the intersection of a parabola and a line, a circle and a line, and a circle and an ellipse. The methods for solving systems of nonlinear equations are similar to those for linear equations.

Solving a System of Nonlinear Equations Using Substitution

A system of nonlinear equations is a system of two or more equations in two or more variables containing at least one equation that is not linear. Recall that a linear equation can take the form [latex]Ax+By+C=0[/latex]. Any equation that cannot be written in this form in nonlinear. The substitution method we used for linear systems is the same method we will use for nonlinear systems. We solve one equation for one variable and then substitute the result into the second equation to solve for another variable, and so on. There is, however, a variation in the possible outcomes.

Intersection of a Parabola and a Line

There are three possible types of solutions for a system of nonlinear equations involving a parabola and a line.

A General Note: Possible Types of Solutions for Points of Intersection of a Parabola and a Line

Figure 2 illustrates possible solution sets for a system of equations involving a parabola and a line.

• No solution. The line will never intersect the parabola.
• One solution. The line is tangent to the parabola and intersects the parabola at exactly one point.
• Two solutions. The line crosses on the inside of the parabola and intersects the parabola at two points.

Figure 2

Example 1: Solving a System of Nonlinear Equations Representing a Parabola and a Line

Solve the system of equations.

[latex]\begin{gathered}x-y=-1 \\ y={x}^{2}+1 \end{gathered}[/latex]

Show Solution

Solve the first equation for [latex]x[/latex] and then substitute the resulting expression into the second equation.

[latex]\begin{array}{cl}x-y=-1& \\ x=y - 1&\text{Solve for }x. \\ \hfill \\ y={x}^{2}+1& \\ y={\left(y - 1\right)}^{2}+1 &\text{Substitute expression for }x.\end{array}[/latex]

Expand the equation and set it equal to zero.

[latex]\begin{align}y&={\left(y - 1\right)}^{2}+1 \\ y&=\left({y}^{2}-2y+1\right)+1 \\ y&={y}^{2}-2y+2 \\ 0&={y}^{2}-3y+2 \\ 0&=\left(y - 2\right)\left(y - 1\right) \end{align}[/latex]

Solving for [latex]y[/latex] gives [latex]y=2[/latex] and [latex]y=1[/latex]. Next, substitute each value for [latex]y[/latex] into the first equation to solve for [latex]x[/latex]. Always substitute the value into the linear equation to check for extraneous solutions.

[latex]\begin{gathered}x-y=-1 \\ x-\left(2\right)=-1 \\ x=1 \\ \hfill \\ x-\left(1\right)=-1 \\ x=0 \end{gathered}[/latex]

The solutions are [latex]\left(1,2\right)[/latex] and [latex]\left(0,1\right),\text{}[/latex] which can be verified by substituting these [latex]\left(x,y\right)[/latex] values into both of the original equations.

Figure 3

Intersection of a Circle and a Line

Just as with a parabola and a line, there are three possible outcomes when solving a system of equations representing a circle and a line.

A General Note: Possible Types of Solutions for the Points of Intersection of a Circle and a Line

Figure 4 illustrates possible solution sets for a system of equations involving a circle and a line.

• No solution. The line does not intersect the circle.
• One solution. The line is tangent to the circle and intersects the circle at exactly one point.
• Two solutions. The line crosses the circle and intersects it at two points.

Figure 4

Example 2: Finding the Intersection of a Circle and a Line by Substitution

Find the intersection of the given circle and the given line by substitution.

[latex]\begin{gathered}{x}^{2}+{y}^{2}=5 \\ y=3x - 5 \end{gathered}[/latex]

Show Solution

One of the equations has already been solved for [latex]y[/latex]. We will substitute [latex]y=3x - 5[/latex] into the equation for the circle.

[latex]\begin{align}{x}^{2}+{\left(3x - 5\right)}^{2}&=5\\ {x}^{2}+9{x}^{2}-30x+25&=5\\ 10{x}^{2}-30x+20&=0\end{align}[/latex]

Now, we factor and solve for [latex]x[/latex].

[latex]\begin{gathered}10\left({x}^{2}-3x+2\right)=0 \\ 10\left(x - 2\right)\left(x - 1\right)=0 \\ x=2 \\ x=1 \end{gathered}[/latex]

Substitute the two x-values into the original linear equation to solve for [latex]y[/latex].

[latex]\begin{align}y&=3\left(2\right)-5 \\ &=1\hfill \\ y&=3\left(1\right)-5 \\ &=-2 \end{align}[/latex]

The line intersects the circle at [latex]\left(2,1\right)[/latex] and [latex]\left(1,-2\right)[/latex], which can be verified by substituting these [latex]\left(x,y\right)[/latex] values into both of the original equations.

Figure 5

Solving a System of Nonlinear Equations Using Elimination

We have seen that substitution is often the preferred method when a system of equations includes a linear equation and a nonlinear equation. However, when both equations in the system have like variables of the second degree, solving them using elimination by addition is often easier than substitution. Generally, elimination is a far simpler method when the system involves only two equations in two variables (a two-by-two system), rather than a three-by-three system, as there are fewer steps. As an example, we will investigate the possible types of solutions when solving a system of equations representing a circle and an ellipse.

A General Note: Possible Types of Solutions for the Points of Intersection of a Circle and an Ellipse

Figure 6 illustrates possible solution sets for a system of equations involving a circle and an ellipse.

• No solution. The circle and ellipse do not intersect. One shape is inside the other or the circle and the ellipse are a distance away from the other.
• One solution. The circle and ellipse are tangent to each other, and intersect at exactly one point.
• Two solutions. The circle and the ellipse intersect at two points.
• Three solutions. The circle and the ellipse intersect at three points.
• Four solutions. The circle and the ellipse intersect at four points.

Figure 6

Example 3: Solving a System of Nonlinear Equations Representing a Circle and an Ellipse

Solve the system of nonlinear equations.

[latex]\begin{array}{cr}\hfill {x}^{2}+{y}^{2}=26& \hfill \left(1\right)\\ \hfill 3{x}^{2}+25{y}^{2}=100& \hfill \left(2\right)\end{array}[/latex]

Show Solution

Let’s begin by multiplying equation (1) by [latex]-3,\text{}[/latex] and adding it to equation (2).

[latex]\begin{align} \left(-3\right)\left({x}^{2}+{y}^{2}\right)&=\left(-3\right)\left(26\right) \\ -3{x}^{2}-3{y}^{2}&=-78 \\ 3{x}^{2}+25{y}^{2}&=100 \\ \hline 22{y}^{2}&=22 \end{align}[/latex]

After we add the two equations together, we solve for [latex]y[/latex].

[latex]\begin{gathered}{y}^{2}=1 \\ y=\pm \sqrt{1}=\pm 1 \end{gathered}[/latex]

Substitute [latex]y=\pm 1[/latex] into one of the equations and solve for [latex]x[/latex].

[latex]\begin{gathered}{x}^{2}+{\left(1\right)}^{2}=26 \\ {x}^{2}+1=26 \\ {x}^{2}=25 \\ x=\pm \sqrt{25}=\pm 5 \\ \hfill \\ {x}^{2}+{\left(-1\right)}^{2}=26 \\ {x}^{2}+1=26 \\ {x}^{2}=25=\pm 5 \end{gathered}[/latex]

There are four solutions: [latex]\left(5,1\right),\left(-5,1\right),\left(5,-1\right),\text{and}\left(-5,-1\right)[/latex].

Figure 7

 Graphing Nonlinear Inequalities and Systems of Nonlinear Inequalities

All of the equations in the systems that we have encountered so far have involved equalities, but we may also encounter systems that involve inequalities. We have already learned to graph linear inequalities by graphing the corresponding equation, and then shading the region represented by the inequality symbol. Now, we will follow similar steps to graph a nonlinear inequality so that we can learn to solve systems of nonlinear inequalities. A nonlinear inequality is an inequality containing a nonlinear expression. Graphing a nonlinear inequality is much like graphing a linear inequality.

Recall that when the inequality is greater than, [latex]y>a[/latex], or less than, [latex]y

Figure 8. (a) an example of [latex]y>a[/latex]; (b) an example of [latex]y\ge a[/latex]; (c) an example of [latex]y<a[/latex]; (d) an example of [latex]y\le a[/latex]