Real-world application problems for two-step equations Grade 7.

📚 What are Two-Step Equations?

Two-step equations are algebraic equations that require two operations (like addition, subtraction, multiplication, or division) to solve for the unknown variable. They bridge the gap between simple one-step equations and more complex multi-step problems. Understanding them is crucial for mastering algebra!

📜 History and Background

The concept of solving equations has ancient roots, dating back to early civilizations like the Egyptians and Babylonians. However, the symbolic algebra we use today developed gradually through the work of mathematicians from various cultures. The formalization of algebraic notation allowed for more efficient and systematic problem-solving, eventually leading to the methods we use for solving equations today.

🔑 Key Principles for Solving

• ⚖️ Isolate the Variable: The main goal is to get the variable (usually represented by letters like $x$ or $y$) by itself on one side of the equation.
• ➕ Undo Addition/Subtraction First: If there's addition or subtraction in the equation, perform the inverse operation to cancel it out. For example, if the equation has '+ 3', subtract 3 from both sides.
• ➗ Undo Multiplication/Division Second: After dealing with addition or subtraction, address any multiplication or division. Perform the inverse operation to isolate the variable further.
• 🔄 Keep it Balanced: Whatever operation you perform on one side of the equation, you MUST perform the same operation on the other side to maintain equality. This is the golden rule of equation solving!

🍕 Real-World Examples

Let's look at some scenarios where you can use two-step equations:

• 📦 Shipping Costs: A company charges a \$5 flat fee for shipping plus \$2 per item. If your total bill is \$13, how many items did you order?
• Let $x$ be the number of items ordered. The equation is $2x + 5 = 13$.
• Subtract 5 from both sides: $2x = 8$.
• Divide both sides by 2: $x = 4$. You ordered 4 items.
• 🎫 Movie Tickets: You and two friends go to the movies. You have a coupon for \$3 off the total. If you paid \$24, how much was each ticket?
• Let $t$ be the cost of one ticket. The equation is $3t - 3 = 24$.
• Add 3 to both sides: $3t = 27$.
• Divide both sides by 3: $t = 9$. Each ticket cost \$9.
• 🎮 Arcade Games: An arcade charges \$2 to enter and \$0.50 per game. If you spent \$7, how many games did you play?
• Let $g$ be the number of games played. The equation is $0.50g + 2 = 7$.
• Subtract 2 from both sides: $0.50g = 5$.
• Divide both sides by 0.50: $g = 10$. You played 10 games.
• 🌡️ Temperature Conversion: The formula to convert Celsius to Fahrenheit is $F = \frac{9}{5}C + 32$. If the temperature is 77°F, what is it in Celsius?
• The equation is $\frac{9}{5}C + 32 = 77$.
• Subtract 32 from both sides: $\frac{9}{5}C = 45$.
• Multiply both sides by $\frac{5}{9}$: $C = 25$. The temperature is 25°C.
• 📱 Phone Bill: Your phone plan costs \$20 per month plus \$0.10 per text message. If your bill was \$25, how many texts did you send?
• Let $m$ be the number of text messages. The equation is $0.10m + 20 = 25$.
• Subtract 20 from both sides: $0.10m = 5$.
• Divide both sides by 0.10: $m = 50$. You sent 50 text messages.
• 🧁 Baking Cookies: You need 2 cups of flour for the recipe, and then you want to add an extra \(\frac{1}{4}\) cup of flour per cookie. If you use a total of 4 cups of flour, how many cookies did you make?
• Let $c$ be the number of cookies made. The equation is $\frac{1}{4}c + 2 = 4$.
• Subtract 2 from both sides: $\frac{1}{4}c = 2$.
• Multiply both sides by 4: $c = 8$. You made 8 cookies.
• 💰 Saving Money: You start with \$10 and save \$3 each week. After how many weeks will you have \$46?
• Let $w$ be the number of weeks. The equation is $3w + 10 = 46$.
• Subtract 10 from both sides: $3w = 36$.
• Divide both sides by 3: $w = 12$. It will take 12 weeks.

📝 Practice Quiz

Test your understanding with these problems:

1. Sarah buys 3 apples and a \$2 drink. She spends \$5 in total. How much does each apple cost?
2. John earns \$8 per hour and receives a \$15 bonus. He earns \$79 in total. How many hours did he work?
3. A taxi charges \$3 plus \$0.50 per mile. The total fare is \$8. How many miles were driven?

Answers:

1. \$1
2. 8 hours
3. 10 miles

💡 Conclusion

Two-step equations are powerful tools for solving real-world problems. By understanding the principles of isolating the variable and performing inverse operations, you can confidently tackle various scenarios. Keep practicing, and you'll become a master equation solver! 🌟