7.3 Systems of nonlinear equations and inequalities: two variables  (Page 3/9)

Solving a system of nonlinear equations representing a circle and an ellipse

Solve the system of nonlinear equations.

$$\begin{array}{rr}\hfill x^2+y^2=26& \hfill (1)\\ \hfill 3x^2+25y^2=100& \hfill (2)\end{array}$$

Let’s begin by multiplying equation (1) by $\mathrm{-3},$ and adding it to equation (2).

$$\frac{\begin{array}{l}\begin{array}{l}\hfill \\ (-3)(x^2+y^2)=(-3)(26)\hfill \\ -3x^2-3y^2=-78\hfill \end{array}\hfill \\ 3x^2+25y^2=100\hfill \end{array}}{22y^2=22}$$

After we add the two equations together, we solve for $y.$

$$\begin{array}{l}{y}^2=1\hfill \\ y=\pm \sqrt{1}=\pm 1\hfill \end{array}$$

Substitute $y=\pm 1$ into one of the equations and solve for $x.$

$$\begin{array}{l}{x}^2+{(1)}^2=26\hfill \\ x^2+1=26\hfill \\ x^2=25\hfill \\ x=\pm \sqrt{25}=\pm 5\hfill \\ \hfill \\ x^2+{(\mathrm{-1})}^2=26\hfill \\ x^2+1=26\hfill \\ x^2=25=\pm 5\hfill \end{array}$$

There are four solutions: $\left(5,1\right),\left(\mathrm{-5},1\right),\left(5,\mathrm{-1}\right),\text{and}\left(\mathrm{-5},\mathrm{-1}\right).$ See [link] .

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Graphing a nonlinear inequality

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, $y>a,$ or less than, $y<a,$ the graph is drawn with a dashed line. When the inequality is greater than or equal to, $y\ge a,$ or less than or equal to, $y\le a,$ the graph is drawn with a solid line. The graphs will create regions in the plane, and we will test each region for a solution. If one point in the region works, the whole region works. That is the region we shade. See [link] .

Graphing an inequality for a parabola

Graph the inequality $y>x^2+1.$

First, graph the corresponding equation $y=x^2+1.$ Since $y>x^2+1$ has a greater than symbol, we draw the graph with a dashed line. Then we choose points to test both inside and outside the parabola. Let’s test the points
$\left(0,2\right)$ and $\left(2,0\right).$ One point is clearly inside the parabola and the other point is clearly outside.

$$\begin{array}{ll}y>x^2+1\hfill & \hfill \\ 2>{(0)}^2+1\hfill & \hfill \\ 2>1\hfill & \text{True}\hfill \\ \hfill & \hfill \\ \hfill & \hfill \\ \hfill & \hfill \\ 0>{(2)}^2+1\hfill & \hfill \\ 0>5\hfill & \text{False}\hfill \end{array}$$

The graph is shown in [link] . We can see that the solution set consists of all points inside the parabola, but not on the graph itself.

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