Elimination Method for Systems of Equations Involving Conics

📚 Understanding the Elimination Method for Conics

The elimination method is a technique used to solve systems of equations by eliminating one variable. When dealing with systems involving conic sections (circles, ellipses, parabolas, hyperbolas), the process is similar to linear systems but often requires more algebraic manipulation. Here's a comprehensive guide:

📜 History and Background

The concept of solving simultaneous equations dates back to ancient civilizations. However, the systematic approach we know today evolved over centuries. The elimination method, a cornerstone of linear algebra, extends naturally to systems involving conic sections, which have been studied since the time of the ancient Greeks, particularly by Apollonius of Perga.

🔑 Key Principles

• 🎯 Identify the Target Variable: Determine which variable is easiest to eliminate based on the coefficients in the equations.
• ⚖️ Multiply Equations: Multiply one or both equations by constants so that the coefficients of the target variable are opposites.
• ➕ Add the Equations: Add the modified equations together. This should eliminate the target variable.
• ✍️ Solve for Remaining Variable: Solve the resulting equation for the remaining variable.
• 🔄 Substitute Back: Substitute the value(s) found back into one of the original equations to solve for the eliminated variable.
• ✅ Check Solutions: Verify your solutions by substituting them into both original equations.

📝 Step-by-Step Guide

1. 🔢 Write down the system of equations: Ensure both equations are clearly written and organized.
2. 🎯 Choose a variable to eliminate: Look for opportunities where coefficients are the same or easily made the same (but with opposite signs).
3. ✖️ Multiply one or both equations: Do this so that the coefficients of the chosen variable are additive inverses.
4. ➕ Add the equations: This should eliminate one variable.
5. 💡 Solve for the remaining variable: This might involve factoring or using the quadratic formula.
6. ↩️ Substitute back: Plug the value(s) you found back into one of the original equations to solve for the other variable.
7. 🧐 Check your solutions: Make sure the solutions satisfy both original equations.

➗ Example 1: Circle and Line

Solve the system:

$x^2 + y^2 = 25$

$y = x + 1$

1. ➡️ Substitution: Substitute $y$ in the first equation: $x^2 + (x+1)^2 = 25$
2. ⚙️ Simplify: $x^2 + x^2 + 2x + 1 = 25 \Rightarrow 2x^2 + 2x - 24 = 0$
3. ➗ Divide by 2: $x^2 + x - 12 = 0$
4. 💡 Factor: $(x+4)(x-3) = 0 \Rightarrow x = -4, 3$
5. ↩️ Find y: For $x = -4$, $y = -4 + 1 = -3$. For $x = 3$, $y = 3 + 1 = 4$.
6. ✅ Solutions: $(-4, -3)$ and $(3, 4)$

📐 Example 2: Parabola and Line

Solve the system:

$y = x^2 - 4x + 5$

$y = x + 1$

1. ➡️ Substitution: Substitute $y$ in the first equation: $x + 1 = x^2 - 4x + 5$
2. ⚙️ Rearrange: $x^2 - 5x + 4 = 0$
3. 💡 Factor: $(x-4)(x-1) = 0 \Rightarrow x = 4, 1$
4. ↩️ Find y: For $x = 4$, $y = 4 + 1 = 5$. For $x = 1$, $y = 1 + 1 = 2$.
5. ✅ Solutions: $(4, 5)$ and $(1, 2)$

📈 Example 3: Two Conics

Solve the system:

$x^2 + y^2 = 9$

$x^2 - y = 3$

1. ➖ Eliminate $x^2$: Subtract the second equation from the first.
2. ⚙️ Result: $y^2 + y = 6$
3. 💡 Rearrange: $y^2 + y - 6 = 0$
4. 💡 Factor: $(y+3)(y-2) = 0 \Rightarrow y = -3, 2$
5. ↩️ Find x: If $y = -3$, $x^2 = 0 \Rightarrow x = 0$. If $y = 2$, $x^2 = 5 \Rightarrow x = \pm\sqrt{5}$.
6. ✅ Solutions: $(0, -3)$, $(\sqrt{5}, 2)$, $(-\sqrt{5}, 2)$

💡 Tips and Tricks

• 🧐 Look for Easy Eliminations: Sometimes, a variable can be eliminated with minimal manipulation.
• 🔄 Rearrange Equations: If necessary, rearrange equations to align terms for easier elimination.
• ⚠️ Watch for Extraneous Solutions: Always check your solutions in the original equations, especially when dealing with radicals.
• ✍️ Practice: The more you practice, the better you'll become at recognizing patterns and applying the elimination method effectively.

📝 Practice Quiz

Solve the following systems of equations using the elimination method:

1. ❓ $x^2 + y^2 = 16$ and $y = x - 4$
2. ❓ $y = x^2 - 2x + 1$ and $y = x + 1$
3. ❓ $x^2 + y^2 = 25$ and $x + y = 1$
4. ❓ $y = x^2 + 3x - 2$ and $y = 2x + 1$
5. ❓ $x^2 - y = 5$ and $x - y = -3$
6. ❓ $x^2 + y^2 = 4$ and $y = x^2 - 2$
7. ❓ $x^2 + y = 7$ and $x^2 - y = 1$

🔑 Solutions to Practice Quiz

1. ✅ $(4, 0), (0, -4)$
2. ✅ $(0, 1), (3, 4)$
3. ✅ $(4, -3), (-3, 4)$
4. ✅ $(-3, -5), (1, 3)$
5. ✅ $(2, 5), (-1, 2)$
6. ✅ $(0, -2), (\sqrt{3}, 1), (-\sqrt{3}, 1)$
7. ✅ $(2, 3), (-2, 3)$

🌍 Real-World Applications

Systems of equations involving conics appear in various fields:

• 🛰️ Satellite Orbits: Determining the intersection of satellite paths with the Earth's surface.
• 🌉 Engineering: Designing structures with parabolic or elliptical shapes.
• 🎮 Computer Graphics: Calculating intersections in 3D modeling and rendering.

🏁 Conclusion

The elimination method is a powerful tool for solving systems of equations, even when conic sections are involved. By understanding the key principles and practicing regularly, you can master this technique and apply it to various mathematical and real-world problems.