Welcome back to class! This week, we are diving deep into the interaction between different types of functions. In Chapter 11, Sections 6 and 8, we move beyond simple straight lines to explore how curves intersect and how to map out regions of solutions. Whether you represent these systems algebraically or graphically, the logic remains consistent.

Section 11.6: Systems of Nonlinear Equations

In previous sections, we solved systems where all equations were linear. Now, we introduce equations with degrees higher than one (like circles, parabolas, or hyperbolas). A nonlinear system contains at least one nonlinear equation.

We primarily use two methods to solve these:

  • Substitution Method: Ideal when one variable is easy to isolate. You solve for one variable and substitute the expression into the nonlinear equation.
  • Elimination (Addition) Method: Useful when you can align terms to cancel out a variable when the equations are added together.

For example, consider finding the intersection of a circle and a line:

$$ \begin{cases} x^2 + y^2 = 25 \\ x - y = 1 \end{cases} $$

Here, we would solve the second equation for $x$ (so $x = y + 1$) and substitute it into the first equation to solve for $y$. Remember, unlike linear systems, nonlinear systems can have multiple solutions (points of intersection) or no solution at all.

Section 11.8: Systems of Inequalities

While Section 11.6 focuses on specific points of equality, Section 11.8 looks at entire regions of valid solutions. A system of inequalities asks us to find the set of points $(x, y)$ that satisfy all inequalities simultaneously.

Key Steps for Success:

  1. Graph the Boundaries: Treat the inequality like an equation. Use a dashed line for strict inequalities ($<$ or $>$) and a solid line for inclusive inequalities ($\leq$ or $\geq$).
  2. Test Points: Pick a point not on the line (like $(0,0)$) to determine which side of the boundary to shade.
  3. Find the Intersection: The solution to the system is the region where the shading for all inequalities overlaps.

Consider a system involving a parabola and a line:

$$ \begin{cases} y > x^2 \\ x + y < 4 \end{cases} $$

The solution region is the area inside the parabola but below the line $y = -x + 4$.

Study Tip: Be sure to watch the attached lecture video where I walk through graphing these regions by hand. It is crucial to be precise with your boundary lines to see exactly where the solution set lies!