The Basic Steps to Solve Any One-Variable Equation

📚 Understanding One-Variable Equations

A one-variable equation is a mathematical statement that asserts the equality of two expressions, both of which contain only one unknown variable. Solving such an equation means finding the value of the variable that makes the equation true.

📜 Historical Context

The pursuit of solving equations dates back to ancient civilizations. Egyptians and Babylonians tackled linear equations using methods like the 'rule of false position.' Diophantus, a Greek mathematician, further developed algebraic notations and methods in his work 'Arithmetica' during the 3rd century AD. Over centuries, mathematicians refined techniques, leading to the systematic approaches we use today.

🔑 Key Principles

• ⚖️ Equality Preservation: The golden rule is to maintain balance. Whatever operation you perform on one side of the equation, you must perform the same operation on the other side.
• ➕ Addition/Subtraction Principle: You can add or subtract the same number from both sides of the equation without changing the solution. This helps isolate the variable.
• ➗ Multiplication/Division Principle: Similarly, you can multiply or divide both sides of the equation by the same non-zero number. This is crucial for solving equations where the variable is multiplied by a coefficient.
• 🤝 Combining Like Terms: Simplify each side of the equation by combining like terms (constants with constants, variable terms with variable terms).
• 🎯 Inverse Operations: Use inverse operations to isolate the variable. For example, use subtraction to undo addition, and division to undo multiplication.

🪜 Step-by-Step Solution Guide

1. Simplify:
   • 🧹 Clear parentheses by distributing.
   • ➕ Combine like terms on each side of the equation.
2. Isolate the Variable Term:
   • ➕ Use addition or subtraction to move all terms containing the variable to one side of the equation and all constants to the other side.
3. Solve for the Variable:
   • ➗ Use multiplication or division to isolate the variable.
4. Check Your Solution:
   • ✔️ Substitute the value you found back into the original equation to verify that it makes the equation true.

🧮 Example 1: Solving a Basic Linear Equation

Solve for $x$: $3x + 5 = 14$

1. Subtract 5 from both sides: $3x + 5 - 5 = 14 - 5$, which simplifies to $3x = 9$.
2. Divide both sides by 3: $\frac{3x}{3} = \frac{9}{3}$, which gives $x = 3$.
3. Check: $3(3) + 5 = 9 + 5 = 14$. The solution is correct.

📊 Example 2: Solving an Equation with Distribution

Solve for $y$: $2(y - 1) = 6$

1. Distribute the 2: $2y - 2 = 6$.
2. Add 2 to both sides: $2y - 2 + 2 = 6 + 2$, which simplifies to $2y = 8$.
3. Divide both sides by 2: $\frac{2y}{2} = \frac{8}{2}$, which gives $y = 4$.
4. Check: $2(4 - 1) = 2(3) = 6$. The solution is correct.

📈 Example 3: Solving an Equation with Fractions

Solve for $z$: $\frac{z}{4} - 3 = 2$

1. Add 3 to both sides: $\frac{z}{4} - 3 + 3 = 2 + 3$, which simplifies to $\frac{z}{4} = 5$.
2. Multiply both sides by 4: $4 \cdot \frac{z}{4} = 4 \cdot 5$, which gives $z = 20$.
3. Check: $\frac{20}{4} - 3 = 5 - 3 = 2$. The solution is correct.

💡 Tips and Tricks

• 📝 Always Check Your Work: Substitution is your best friend. Ensure your solution satisfies the original equation.
• 🧮 Stay Organized: Keep your work neat and clearly labeled to avoid errors.
• 💪 Practice Regularly: The more you practice, the more comfortable you'll become with different types of equations.

✅ Conclusion

Solving one-variable equations is a fundamental skill in algebra. By understanding the basic principles and following a systematic approach, you can tackle a wide range of problems. Remember to practice regularly and always check your work. Happy solving!