Solved Problems: Two-Step Equations (ax + b = c) for Grade 8

📚 What are Two-Step Equations?

A two-step equation is an algebraic equation that requires two operations to solve for the variable. These equations typically follow the form $ax + b = c$, where $a$, $b$, and $c$ are numbers, and $x$ is the variable you need to find.

📜 A Little History

The concept of solving equations dates back to ancient civilizations, with early forms found in Babylonian and Egyptian mathematics. However, the symbolic notation we use today developed gradually through the work of mathematicians like François Viète in the 16th century. Two-step equations are a fundamental building block in algebra, providing a simple introduction to more complex problem-solving.

🔑 Key Principles for Solving Two-Step Equations

• ➕ Isolate the Term with the Variable: First, use inverse operations (addition or subtraction) to isolate the term containing the variable (e.g., $ax$).
• ➗ Solve for the Variable: Next, use inverse operations (multiplication or division) to solve for the variable itself (e.g., $x$).
• ⚖️ Maintain Balance: Remember to perform the same operation on both sides of the equation to keep it balanced. What you do on one side, you MUST do on the other!
• ✅ Check Your Answer: Substitute your solution back into the original equation to verify that it is correct.

✍️ Solving a Two-Step Equation: A Worked Example

Let's solve the equation $3x + 5 = 14$:

1. Subtract 5 from both sides: $3x + 5 - 5 = 14 - 5$ $3x = 9$
2. Divide both sides by 3: $\frac{3x}{3} = \frac{9}{3}$ $x = 3$

Therefore, the solution is $x = 3$.

🌍 Real-World Examples

• 🍕 Pizza Party: You're buying pizza for a party. Each pizza costs $12, and there's a $5 delivery fee. You spent $41 in total. How many pizzas did you buy? Equation: $12x + 5 = 41$
• 🎬 Movie Tickets: You and a friend go to the movies. Tickets cost the same price each. You also buy a $7 popcorn to share. The total bill was $23. How much did each ticket cost? Equation: $2x + 7 = 23$
• 🏃 Running Laps: You run laps around a track. You run 3 laps, then walk 2 laps. If you ran a total of 17 laps after repeating this several times, how many times did you complete the 3 running laps? Equation: $3x + 2 = 17$

📝 Practice Quiz

Solve the following equations:

1. $2x + 4 = 10$
2. $5y - 3 = 12$
3. $4z + 7 = 19$
4. $6a - 2 = 22$
5. $3b + 1 = 16$
6. $7c - 5 = 9$
7. $8d + 3 = 35$

Answers:

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

💡 Tips for Success

• ✅ Show Your Work: Write down each step clearly to avoid mistakes.
• 🧐 Double-Check: Review your calculations carefully.
• 🤝 Practice Regularly: The more you practice, the easier it becomes!
• 🤯 Don't Give Up: If you get stuck, try a different approach or ask for help.

🎓 Conclusion

Mastering two-step equations is a crucial step in your algebra journey. With practice and a clear understanding of the principles, you'll be solving these problems with ease in no time!