Word problems involving equations with variables on both sides solutions

📚 Understanding Equations with Variables on Both Sides

An equation with variables on both sides is a mathematical statement where the same variable appears on both the left and right sides of the equals sign. The goal is to isolate the variable on one side to find its value. Solving these equations often involves combining like terms and using inverse operations.

📜 A Brief History

The concept of equations and solving for unknowns dates back to ancient civilizations. Egyptians and Babylonians used methods to solve linear equations. The formalization of algebra, which includes manipulating equations with variables, developed further in Islamic mathematics during the Middle Ages and was later refined in Europe.

🔑 Key Principles for Solving

• ⚖️ Maintain Balance: Remember, an equation is like a balance scale. Whatever you do to one side, you must do to the other to keep it balanced.
• ➕ Combine Like Terms: Simplify each side of the equation by combining like terms (e.g., $3x + 2x = 5x$).
• ➖ Isolate the Variable: Use inverse operations (addition, subtraction, multiplication, division) to get the variable term alone on one side of the equation.
• ➗ Solve for the Variable: Perform the necessary operations to find the value of the variable.
• ✔️ Check Your Solution: Substitute the value you found back into the original equation to verify that it makes the equation true.

🌐 Real-World Examples

Example 1: Comparing Phone Plans

Two phone companies offer different plans. Company A charges $20 per month plus $0.10 per minute. Company B charges $30 per month plus $0.05 per minute. How many minutes would you need to use for the plans to cost the same?

Solution:

Let $m$ represent the number of minutes.

Company A's cost: $20 + 0.10m$

Company B's cost: $30 + 0.05m$

Set the costs equal to each other: $20 + 0.10m = 30 + 0.05m$

Subtract $0.05m$ from both sides: $20 + 0.05m = 30$

Subtract $20$ from both sides: $0.05m = 10$

Divide both sides by $0.05$: $m = 200$

So, the plans would cost the same if you used 200 minutes.

Example 2: Balancing a Budget

You are saving money for a new bicycle. You currently have $50 saved, and you save $10 per week. Your friend has $80 saved, and they save $5 per week. After how many weeks will you both have the same amount of money?

Solution:

Let $w$ represent the number of weeks.

Your savings: $50 + 10w$

Friend's savings: $80 + 5w$

Set the savings equal to each other: $50 + 10w = 80 + 5w$

Subtract $5w$ from both sides: $50 + 5w = 80$

Subtract $50$ from both sides: $5w = 30$

Divide both sides by $5$: $w = 6$

So, after 6 weeks, you and your friend will have the same amount of money.

✍️ Practice Quiz

1. ❓ Solve for $x$: $3x + 5 = x - 1$
2. 🧮 Solve for $y$: $7y - 3 = 4y + 9$
3. 🤔 Solve for $a$: $6a + 2 = -2a - 14$
4. 📈 Solve for $b$: $5b - 8 = -3b + 8$
5. 💡 Solve for $z$: $4z + 7 = -z - 3$
6. ➗ Solve for $p$: $2p - 6 = -4p + 12$
7. ➕ Solve for $q$: $9q + 1 = -5q - 69$

✅ Conclusion

Mastering equations with variables on both sides involves understanding the basic principles of algebra and applying them systematically. By balancing equations and using inverse operations, you can confidently solve these problems and apply them to various real-world scenarios. Practice consistently, and you'll find these equations become much less daunting!