what are linear equations in one variable?

📚 What are Linear Equations in One Variable?

A linear equation in one variable is an equation that can be written in the form $ax + b = 0$, where $x$ represents the variable, and $a$ and $b$ are constants (numbers), with $a \neq 0$. The key characteristic is that the highest power of the variable $x$ is 1. This means there are no $x^2$, $x^3$, or higher-order terms. The solution to a linear equation is the value of the variable that makes the equation true.

📜 A Brief History

The concept of solving equations dates back to ancient civilizations. Egyptians and Babylonians were solving linear equations as far back as 2000 BC, though their methods and notations were quite different from what we use today. The development of algebraic notation, especially by mathematicians like Muhammad al-Khwarizmi in the 9th century, significantly streamlined the solving of these equations. Al-Khwarizmi is often considered the 'father of algebra' due to his influential work.

🔑 Key Principles for Solving Linear Equations

• ⚖️ The Golden Rule: Whatever you do to one side of the equation, you must do to the other side to maintain equality.
• ➕ Addition Property of Equality: You can add the same number to both sides of an equation without changing its solution.
• ➖ Subtraction Property of Equality: You can subtract the same number from both sides of an equation without changing its solution.
• ➗ Division Property of Equality: You can divide both sides of an equation by the same non-zero number without changing its solution.
• ✖️ Multiplication Property of Equality: You can multiply both sides of an equation by the same number without changing its solution.
• 🤝 Simplification: Before applying the properties of equality, simplify each side of the equation by combining like terms.
• 🎯 Isolate the Variable: The ultimate goal is to isolate the variable on one side of the equation.

🌍 Real-world Examples

Linear equations pop up everywhere! Here are a few examples:

• 💰 Budgeting: If you earn $\$15$ per hour and need to save $\$300$ for a new phone, the equation $15x = 300$ represents how many hours ($x$) you need to work.
• 🚶 Travel: If you're walking at a constant speed of $3$ miles per hour, the equation $d = 3t$ relates the distance ($d$) you travel to the time ($t$) spent walking.
• 🌡️ Temperature Conversion: The equation $F = (9/5)C + 32$ converts Celsius ($C$) to Fahrenheit ($F$). If you know the Fahrenheit temperature, you can solve for Celsius using a linear equation.

✍️ Practice Quiz

Solve the following linear equations:

1. $2x + 5 = 11$
2. $3y - 7 = 8$
3. $-4z + 2 = -10$
4. $\frac{1}{2}a - 3 = 1$
5. $5b + 9 = 4b - 3$
6. $6c - 4 = 2c + 8$
7. $7 - 2d = 15$

Answers:

1. $x = 3$
2. $y = 5$
3. $z = 3$
4. $a = 8$
5. $b = -12$
6. $c = 3$
7. $d = -4$

💡 Conclusion

Linear equations in one variable are a fundamental concept in algebra. Understanding them provides a solid foundation for more advanced mathematical topics. By grasping the key principles and practicing regularly, anyone can master solving these equations.