Using substitution to solve systems of two quadratic equations

📚 Introduction to Solving Quadratic Systems with Substitution

Solving systems of equations involving quadratics can seem daunting, but the method of substitution provides a powerful and elegant approach. This guide will provide a comprehensive overview, from the foundational principles to practical examples.

📜 Historical Context

The study of quadratic equations and systems dates back to ancient civilizations. Babylonian mathematicians were solving quadratic equations as early as 1800 BC. The development of algebraic notation over centuries eventually led to the systematic methods we use today, including substitution, for solving these equations.

• 🏺 Ancient Roots: Quadratic problems appear on Babylonian clay tablets.
• 📈 Algebraic Evolution: Development of symbolic notation facilitated complex problem-solving.
• 🧑‍🏫 Modern Methods: Substitution emerged as a key technique in solving systems.

🔑 Key Principles of Substitution

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation in one variable, which can then be solved.

• 🎯 Isolate a Variable: Choose the easiest equation to solve for one variable.
• 🔄 Substitute: Replace the isolated variable in the other equation.
• 🧩 Solve the Equation: Solve the resulting single-variable equation.
• 🔙 Back-Substitute: Substitute the solution back to find the other variable's value.
• ✅ Check Your Solution: Verify the solution satisfies both original equations.

💡 Step-by-Step Guide

Here's a detailed breakdown of how to use substitution to solve systems of two quadratic equations:

1. Step 1: Choose an Equation and Variable: Select the equation that looks easiest to manipulate. Solve for one variable in terms of the other.
2. Step 2: Perform the Substitution: Substitute the expression obtained in Step 1 into the other equation.
3. Step 3: Solve the Resulting Equation: You should now have a single equation with only one variable. Solve for that variable. This might involve factoring, using the quadratic formula, or other algebraic techniques.
4. Step 4: Back-Substitution: Once you have the value(s) of one variable, substitute those values back into the equation you used in Step 1 to find the corresponding value(s) of the other variable.
5. Step 5: Check Your Solution(s): Always check your solution(s) by plugging them back into both of the original equations to ensure they satisfy both.

➗ Example 1: A Simple Case

Consider the system: $y = x^2 - 3$ $y = x - 1$

• 🔍 Isolate: Already done! Both equations are solved for $y$.
• 🔄 Substitute: Substitute $x - 1$ for $y$ in the first equation: $x - 1 = x^2 - 3$.
• 🧩 Solve: Rearrange to get $x^2 - x - 2 = 0$. Factoring gives $(x - 2)(x + 1) = 0$, so $x = 2$ or $x = -1$.
• 🔙 Back-Substitute: If $x = 2$, $y = 2 - 1 = 1$. If $x = -1$, $y = -1 - 1 = -2$.
• ✅ Check: $(2, 1)$ and $(-1, -2)$ both satisfy the original equations.

➕ Example 2: A More Complex Scenario

Consider the system: $y = x^2 + 2x - 1$ $2x + y = 2$

• 🎯 Isolate: Solve the second equation for $y$: $y = 2 - 2x$.
• 🔄 Substitute: Substitute $2 - 2x$ for $y$ in the first equation: $2 - 2x = x^2 + 2x - 1$.
• ➗ Solve: Rearrange to get $x^2 + 4x - 3 = 0$. Use the quadratic formula: $x = \frac{-4 \pm \sqrt{4^2 - 4(1)(-3)}}{2(1)} = \frac{-4 \pm \sqrt{28}}{2} = -2 \pm \sqrt{7}$.
• 🧪 Back-Substitute: If $x = -2 + \sqrt{7}$, $y = 2 - 2(-2 + \sqrt{7}) = 6 - 2\sqrt{7}$. If $x = -2 - \sqrt{7}$, $y = 2 - 2(-2 - \sqrt{7}) = 6 + 2\sqrt{7}$.
• 📝 Solutions: The solutions are $(-2 + \sqrt{7}, 6 - 2\sqrt{7})$ and $(-2 - \sqrt{7}, 6 + 2\sqrt{7})$.

🌍 Real-World Applications

Solving systems of quadratic equations has applications in various fields:

• 🛰️ Physics: Projectile motion analysis.
• 🌉 Engineering: Design of curved structures, such as bridges.
• 📈 Economics: Modeling supply and demand curves.

💡 Tips and Tricks

• 🔑 Choose Wisely: Select the simplest equation to solve for a variable.
• 📉 Be Careful with Signs: Pay close attention to signs when substituting.
• 🔬 Double-Check: Always verify your solutions in the original equations.

📝 Practice Quiz

Solve the following systems of equations using substitution:

1. $y = x^2$, $y = 2x + 3$
2. $y = x^2 - 4x + 3$, $y = x - 1$
3. $y = -x^2 + 5$, $y = x + 3$

✅ Conclusion

Substitution is a powerful method for solving systems of two quadratic equations. By understanding the key principles and practicing with examples, you can master this technique and apply it to various real-world problems.