Lesson 11.3: Systems of Nonlinear Equations and Inequalities: Two Variables

New Concept

We'll solve systems with at least one non-linear equation, like a parabola or circle. You'll use substitution and elimination to find where these curves intersect and learn to graph the solution regions for systems of nonlinear inequalities.

What’s next

Next, you'll tackle interactive examples showing how to solve these systems with substitution and elimination, then apply your skills in practice problems.

Property

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. The substitution method is used to solve systems with a linear equation. There are three possible types of solutions for a system 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 it at exactly one point.
• Two solutions. The line crosses on the inside of the parabola and intersects it at two points.

To solve, first solve the linear equation for one variable. Then, substitute the expression into the parabola equation and solve for the remaining variable. Finally, check your solutions in both equations.

Examples

• To solve the system $y = x^2 - 3$ and $y = x - 1$, substitute $x - 1$ for $y$ in the first equation. This gives $x - 1 = x^2 - 3$, which simplifies to $x^2 - x - 2 = 0$. Factoring gives $(x-2)(x+1)=0$, so the solutions are $(2, 1)$ and $(-1, -2)$.

Property

When solving a system of equations representing a circle and a line, there are three possible outcomes:

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

To find the solution, solve the linear equation for one variable. Substitute this expression into the circle's equation, solve for the remaining variable, and check your solutions in both equations.

Examples

• To solve $x^2 + y^2 = 13$ and $y = x - 1$, substitute $y$ in the circle equation: $x^2 + (x-1)^2 = 13$. This becomes $2x^2 - 2x - 12 = 0$, or $x^2 - x - 6 = 0$. Factoring gives $(x-3)(x+2)=0$. The solutions are $(3, 2)$ and $(-2, -3)$.

Property

When both equations in a system have like variables of the second degree (e.g., $x^2$ and $y^2$), solving using elimination by addition is often easier than substitution. For a circle and an ellipse, there can be zero, one, two, three, or four solutions.

To solve, multiply one or both equations by a constant to make the coefficients of one variable opposites. Add the equations to eliminate that variable, solve for the remaining variable, and then substitute back to find the values of the eliminated variable.

Examples

• To solve $x^2 + y^2 = 10$ and $2x^2 - y^2 = 17$, add the two equations to get $3x^2 = 27$, so $x = \pm 3$. Substituting $x^2=9$ into the first equation gives $9 + y^2 = 10$, so $y = ± 1$. The four solutions are $(3, 1), (3, -1), (-3, 1),$ and $(-3, -1)$.

Property

A nonlinear inequality is an inequality containing a nonlinear expression. To graph one:

1. Graph the parabola, circle, or other curve as if it were an equation. This is the boundary.
2. If the operator is $\leq$ or $\geq$, use a solid line because the boundary is included in the solution.
3. If the operator is $<$ or $>$, use a dashed line because the boundary is not included.
4. Test a point in one of the regions. If the statement is true, shade the region including the point. If false, shade the other region.

Examples

• To graph $y < x^2 - 2$, draw the parabola $y = x^2 - 2$ with a dashed line. Test the point $(0, 0)$. Since $0 < 0^2 - 2$ is false, shade the region that does not contain $(0, 0)$, which is the area outside the parabola.

• To graph $x^2 + y^2 ≤ 9$, draw the circle $x^2 + y^2 = 9$ with a solid line. Test the point $(0, 0)$. Since $0^2 + 0^2 ≤ 9$ is true, shade the region inside the circle.

Property

A system of nonlinear inequalities is a set of two or more inequalities where at least one is not linear. The solution is the region where the shaded areas of each inequality overlap, known as the feasible region.

To graph the system:

1. Find the intersection points by solving the corresponding system of equations.
2. Graph each nonlinear inequality, shading the solution region for each.
3. Identify the feasible region, which is the intersection of all shaded areas.

Examples

• For the system $y ≥ x^2$ and $y ≤ x+2$, the feasible region is the area on or above the parabola $y=x^2$ and on or below the line $y=x+2$. The solution is the area enclosed between the two curves.