Defining two-step equations involving rational numbers for Grade 7

📚 Defining Two-Step Equations Involving Rational Numbers

A two-step equation is an algebraic equation that requires two operations to solve it. These operations typically involve a combination of addition or subtraction, and multiplication or division. When rational numbers (fractions, decimals, and integers) are included, the equations might appear more complex, but the underlying principles remain the same.

📜 History and Background

The concept of solving equations has ancient roots, dating back to early civilizations in Mesopotamia and Egypt. However, the symbolic notation we use today largely developed from the 16th century onwards. Rational numbers have always been integral to mathematical problem-solving, and their inclusion in algebraic equations reflects real-world applications where quantities are not always whole numbers.

🔑 Key Principles

• ⚖️ Inverse Operations: Use inverse operations to isolate the variable. For example, if the equation involves addition, use subtraction to undo it.
• ➗ Order of Operations (Reverse): Follow the reverse order of operations (PEMDAS/BODMAS) to solve the equation. Address addition/subtraction before multiplication/division.
• ➕ Combining Like Terms: Simplify each side of the equation by combining like terms before isolating the variable.
• 🔢 Rational Number Operations: Remember the rules for adding, subtracting, multiplying, and dividing rational numbers (fractions and decimals).
• ✔️ Verification: Always check your solution by substituting it back into the original equation to ensure it holds true.

✍️ Solving Two-Step Equations with Rational Numbers: Examples

Example 1: Fractions

Solve for $x$: $\frac{2}{3}x + \frac{1}{2} = \frac{5}{6}$

1. Subtract $\frac{1}{2}$ from both sides: $\frac{2}{3}x = \frac{5}{6} - \frac{1}{2} = \frac{5}{6} - \frac{3}{6} = \frac{2}{6} = \frac{1}{3}$
2. Multiply both sides by $\frac{3}{2}$: $x = \frac{1}{3} \cdot \frac{3}{2} = \frac{1}{2}$

Example 2: Decimals

Solve for $y$: $0.5y - 1.2 = 0.8$

1. Add 1.2 to both sides: $0.5y = 0.8 + 1.2 = 2.0$
2. Divide both sides by 0.5: $y = \frac{2.0}{0.5} = 4$

Example 3: Mixed Rational Numbers

Solve for $z$: $\frac{3}{4}z + 2.5 = 5.5 $

1. Subtract 2.5 from both sides: $\frac{3}{4}z = 5.5 - 2.5 = 3$
2. Multiply both sides by $\frac{4}{3}$: $z = 3 \cdot \frac{4}{3} = 4$

🌍 Real-World Examples

• 🍕 Pizza Sharing: If you have $\frac{2}{5}$ of a pizza left, and you eat $\frac{1}{4}$ of the original whole pizza, how much of the pizza did you start with? Equation: $\frac{2}{5}x - \frac{1}{4} = 0$, where $x$ is the original amount.
• ⛽ Fuel Calculation: A car has used 2.5 gallons of fuel and now has 10.5 gallons left. If the tank was initially $\frac{3}{4}$ full, what is the tank's capacity? Equation: $\frac{3}{4}x - 2.5 = 10.5$, where $x$ is the tank capacity.
• 🧵 Fabric Measurement: A tailor used 1.75 meters of fabric from a roll. If $\frac{1}{3}$ of the roll remains, how much fabric was originally on the roll? Equation: $\frac{2}{3}x + 1.75 = x$, where $x$ is the original length.

💡 Conclusion

Mastering two-step equations involving rational numbers is fundamental for success in algebra and higher-level mathematics. By understanding the key principles and practicing with real-world examples, students can build confidence and proficiency in solving these equations. Remember to always check your work and apply these skills to practical situations to reinforce your understanding. Keep practicing, and you'll become a pro in no time! 👍