What is a 3x3 Linear System and How to Solve It by Substitution?

📚 What is a 3x3 Linear System?

A 3x3 linear system is a set of three linear equations, each containing three variables (typically $x$, $y$, and $z$). The goal is to find values for these variables that satisfy all three equations simultaneously. These systems often arise in various fields like engineering, physics, and economics.

📜 A Brief History

The study of linear systems dates back to ancient times, with early examples found in Babylonian mathematics. However, the systematic methods we use today evolved primarily in the 18th and 19th centuries, with contributions from mathematicians like Gauss and Jordan. The development of matrix algebra further streamlined the process of solving these systems.

🔑 Key Principles of Solving by Substitution

• 🎯 Isolate a Variable: Choose one equation and solve for one variable in terms of the other two.
• 🔄 Substitute: Substitute the expression you found in step 1 into the other two equations. This will leave you with a 2x2 system.
• 🧩 Solve the 2x2 System: Use substitution or elimination to solve the resulting 2x2 system for the remaining two variables.
• ↩️ Back-Substitute: Substitute the values you found back into one of the original equations (or the expression from step 1) to solve for the third variable.
• ✅ Verify: Check your solution by plugging the values of $x$, $y$, and $z$ into all three original equations to ensure they are satisfied.

🧑‍🏫 Step-by-Step Example

Let's solve the following system:

Equation 1: $x + y + z = 6$

Equation 2: $2x - y + z = 3$

Equation 3: $x + 2y - z = 2$

1. Isolate a Variable: From Equation 1, we can easily isolate $x$: $x = 6 - y - z$
2. Substitute: Substitute this expression for $x$ into Equations 2 and 3:
• Equation 2 becomes: $2(6 - y - z) - y + z = 3 \Rightarrow 12 - 2y - 2z - y + z = 3 \Rightarrow -3y - z = -9$
• Equation 3 becomes: $(6 - y - z) + 2y - z = 2 \Rightarrow 6 + y - 2z = 2 \Rightarrow y - 2z = -4$
3. Solve the 2x2 System: Now we have a 2x2 system:
   • $-3y - z = -9$
   • $y - 2z = -4$
   Solve the second equation for $y$: $y = 2z - 4$. Substitute into the first equation:
   • $-3(2z - 4) - z = -9 \Rightarrow -6z + 12 - z = -9 \Rightarrow -7z = -21 \Rightarrow z = 3$
   Now, substitute $z = 3$ back into $y = 2z - 4$: $y = 2(3) - 4 = 2$
4. Back-Substitute: Substitute $y = 2$ and $z = 3$ back into $x = 6 - y - z$: $x = 6 - 2 - 3 = 1$
5. Solution: The solution is $x = 1$, $y = 2$, and $z = 3$.

➕ Real-World Applications

• 🌍 Geography: Determining the flow rates in interconnected river systems.
• 🧪 Chemistry: Balancing chemical equations.
• ⚙️ Engineering: Analyzing forces in structural frameworks.
• 📈 Economics: Modeling supply and demand in multiple markets.

💡 Tips and Tricks

• 🔢 Choose Wisely: When isolating a variable, pick the equation and variable that will result in the simplest expression.
• 🧐 Stay Organized: Keep track of your substitutions to avoid errors.
• 💻 Use Technology: Utilize calculators or software to check your work, especially for complex systems.

📝 Conclusion

Solving 3x3 linear systems by substitution can seem daunting, but by breaking it down into manageable steps and staying organized, you can master this skill. Remember to practice and utilize the tips provided to enhance your problem-solving abilities. Happy solving!