➕ Definition of One-Step Equations
A one-step equation is an algebraic equation that can be solved in only one step. These equations involve only one operation—addition, subtraction, multiplication, or division—performed on a variable.
📜 History and Background
The concept of solving equations dates back to ancient civilizations. Egyptians and Babylonians were solving linear equations thousands of years ago, although their methods were different from modern algebraic notation. The development of symbolic algebra, particularly by mathematicians in the Islamic world during the Middle Ages, paved the way for the systematic approach we use today.
🔑 Key Principles for Solving One-Step Equations
- ⚖️The Golden Rule: What you do to one side of the equation, you must do to the other. This maintains the equality.
- ➕Addition Property: If $a = b$, then $a + c = b + c$. Adding the same number to both sides doesn't change the solution.
- ➖Subtraction Property: If $a = b$, then $a - c = b - c$. Subtracting the same number from both sides doesn't change the solution.
- ✖️Multiplication Property: If $a = b$, then $ac = bc$. Multiplying both sides by the same number doesn't change the solution.
- ➗Division Property: If $a = b$, and $c \neq 0$, then $\frac{a}{c} = \frac{b}{c}$. Dividing both sides by the same non-zero number doesn't change the solution.
- 🎯Inverse Operations: Use the inverse operation to isolate the variable. Addition undoes subtraction, and multiplication undoes division.
➗ Solving Equations with Division
To solve an equation like $\frac{x}{5} = 3$, multiply both sides by 5:
$\frac{x}{5} \times 5 = 3 \times 5$
$x = 15$
✖️ Solving Equations with Multiplication
To solve an equation like $4x = 20$, divide both sides by 4:
$\frac{4x}{4} = \frac{20}{4}$
$x = 5$
➕ Solving Equations with Addition
To solve an equation like $x - 7 = 10$, add 7 to both sides:
$x - 7 + 7 = 10 + 7$
$x = 17$
➖ Solving Equations with Subtraction
To solve an equation like $x + 3 = 8$, subtract 3 from both sides:
$x + 3 - 3 = 8 - 3$
$x = 5$
🌍 Real-World Examples
- 🍕 Splitting the Bill: If a pizza costs $24 and you're splitting it among 4 friends, the equation is $4x = 24$, where $x$ is each person's share. Solving gives $x = $6.
- 🏃 Distance and Speed: If you travel 150 miles in 3 hours at a constant speed, the equation is $3x = 150$, where $x$ is the speed in miles per hour. Solving gives $x = 50$ mph.
- 💰 Saving Money: If you save $15 each week and want to save $135, the equation is $15x = 135$, where $x$ is the number of weeks. Solving gives $x = 9$ weeks.
💡 Tips and Tricks
- ✅ Check Your Work: Always substitute your solution back into the original equation to verify it.
- 📝 Show Your Steps: Write down each step to avoid errors.
- 🧮 Stay Organized: Keep your work neat and tidy to make it easier to follow.
✔️ Conclusion
One-step equations are fundamental to algebra and problem-solving. By understanding the basic principles and practicing regularly, anyone can master them. These equations serve as building blocks for more complex mathematical concepts, making them an essential skill for students and professionals alike.