🌟 Level 3 - Topic 6: Systems of Nonlinear Equations 🚀

1) Understanding Systems of Nonlinear Equations 🤔

A system of nonlinear equations consists of two or more equations, where at least one equation is not linear. This means the system may include quadratic, exponential, absolute value, or other nonlinear functions.

• In Level 2, we learned about systems of linear equations (straight lines).
• In Level 3, we now look at systems where at least one equation is a curve (parabola, circle, etc.).

Example of a Nonlinear System:

$y=x^2-4$

$y=x+2$

One equation is quadratic; the other is linear. The solutions are the points where their graphs intersect.

Number of solutions depends on how the graphs intersect:

• ✅ One solution = one intersection point
• ✅ Two solutions = two intersection points
• ❌ No solution = curves never meet

2) Types of Nonlinear Systems 🌀

2.1 Quadratic-Linear Systems

• One equation is quadratic (parabola)
• The other is linear (straight line)
• Solutions: intersection(s) of line & parabola

Example:

$y=x^2-4$    and    $y=2x$

2.2 Quadratic-Quadratic Systems

• Both equations are quadratic
• Solutions: intersection(s) of two parabolas

Example:

$y=x^2-3$    and    $y=-x^2+5$

2.3 Circle-Linear Systems

• One equation is a circle
• The other is a line
• Solutions: intersection(s) where line touches/crosses the circle

Example:

$x^2+y^2=25$    and    $y=x+3$

3) Methods for Solving Systems of Nonlinear Equations ⚙️

3.1 Substitution Method

1. Solve one equation for a variable
2. Substitute into the other

Example: Solve

$y=x^2-4$    and    $y=2x+1$

1. Step 1: Set equal (both = $y$):

   $x^2-4=2x+1$

2. Step 2: Move all terms to one side:

   $x^2-2x-5=0$

3. Step 3: Apply the quadratic formula:

   $x=\frac{-(-2)\pm \sqrt{(-2{)}^2-4(1)(-5)}}{2\cdot 1}=\frac{2\pm \sqrt{24}}{2}=1\pm \sqrt{6}$

4. Step 4: Substitute each $x$ into $y=2x+1$.

Answer: Two solutions $(1+\sqrt{6},3+2\sqrt{6})$ and $(1-\sqrt{6},3-2\sqrt{6})$.

3.2 Elimination Method

- Best if equations have similar or manipulable terms

Example: Solve

$x^2+y^2=25$    and    $y=x+3$

1. Step 1: Substitute $y=x+3$ into $x^2+y^2=25$:

   $x^2+(x+3{)}^2=25$

2. Step 2: Expand & solve for $x$.
3. Step 3: Find $y$ by substituting $x$.

Answer: Intersection points of the circle & line.

4) Real-Life Applications of Nonlinear Systems 🌍

• Physics: projectile motion (quadratic)
• Economics: supply-demand can be nonlinear
• Engineering: circuits, parabolic reflectors
• Astronomy: orbits are often ellipses/quadratics

Example: A ball’s height:

$h(t)=-5t^2+20t+30$

Find when $h(t)=40$. Solve that nonlinear equation for $t$.

5) Practice Questions 🎯

Fundamental – Build Skills

1. Substitution:

   $y=x^2+2x-3$    and    $y=3x+1$

2. Elimination:

   $x^2+y^2=10$    and    $y=2x$

3. Intersection points:

   $y=-x^2+6x-8$    and    $y=2x-3$

4. Solve for $x,y$:

   $x^2+y^2=25$    and    $y=x+4$

5. Find solutions:

   $x^2-y=4$    and    $x+y=6$

6. Solve for $x,y$:

   $x^2+y^2=9$    and    $x^2-y=1$

7. System with absolute value:

   $y=|x-2|$    and    $y=x^2-4x+3$

Challenging – Push Limits

1. Find max/min in:

   $y=-x^2+4x+5$    and    $y=2x+3$

2. Solve intersection:

   $x^2+y^2=20$    and    $y=2x+1$

3. Parabola eqn through intersections of:

   $y=x^2-6x+9$    and    $y=-2x+8$

4. Ball follows $h(t)=-4.9t^2+20t$, person on 10m building:

   When is the ball at the same height as the building?

5. Two radio towers:

   $(x-3{)}^2+(y-5{)}^2=16$    and    $(x+2{)}^2+(y-1{)}^2=25$

   Find the overlapping points.

6) Summary

• Nonlinear systems: quadratics, circles, exponentials, etc.
• Use substitution or elimination.
• Graphing helps visualize intersection solutions.

🚀 Great work! Keep going with more advanced topics!