Printable Activities: Solving Systems by Substitution (Variable Isolated) for Grade 8

📚 Understanding Systems of Equations

A system of equations is a set of two or more equations containing the same variables. Solving a system means finding values for the variables that make all equations true simultaneously. When using substitution, we leverage the fact that one variable is already isolated (by itself) in one of the equations. This makes it easy to 'plug' that variable's equivalent expression into the other equation.

📜 A Brief History of Substitution

The concept of substitution dates back to ancient mathematical practices. While not explicitly formalized as 'substitution' in modern terms, mathematicians in various civilizations used similar approaches to solve problems involving multiple unknowns. The formalization of algebraic techniques, including substitution, evolved over centuries, solidifying its place as a fundamental tool in algebra.

🔑 Key Principles of Substitution (Variable Isolated)

• 🎯 Identify the Isolated Variable:
Ensure one equation is in the form of $y = ...$ or $x = ...$. This is your starting point.
• 🔌 Substitute:
Replace the isolated variable in the other equation with its equivalent expression.
• ➗ Solve:
Solve the resulting equation for the remaining variable. You'll now have a numerical value for one variable.
• 🔙 Back-Substitute:
Plug the value you just found back into either of the original equations to solve for the other variable.
• ✅ Check:
Verify your solution by substituting both values into both original equations to ensure they hold true.

✍️ Example Problem Walkthrough

Let's solve the following system of equations:

Equation 1: $y = 2x + 1$

Equation 2: $3x + y = 11$

1. Step 1: $y$ is already isolated in Equation 1.
2. Step 2: Substitute $2x + 1$ for $y$ in Equation 2: $3x + (2x + 1) = 11$
3. Step 3: Simplify and solve for $x$: $5x + 1 = 11 \Rightarrow 5x = 10 \Rightarrow x = 2$
4. Step 4: Substitute $x = 2$ back into Equation 1: $y = 2(2) + 1 \Rightarrow y = 5$
5. Step 5: Check the solution $(2, 5)$:
   • Equation 1: $5 = 2(2) + 1$ (True)
   • Equation 2: $3(2) + 5 = 11$ (True)

🌐 Real-World Applications

• 💰 Finance:
Determining break-even points for costs and revenue.
• 🚀 Physics:
Analyzing motion and forces in systems.
• 🌡️ Chemistry:
Calculating concentrations in solutions.
• 🍕 Everyday Life:
Comparing costs of different deals (e.g., pizza sizes vs. price).

📝 Practice Quiz

Solve the following systems of equations using substitution:

1. $y = x + 3$

   $2x + y = 9$

2. $x = 2y - 1$

   $x + 3y = 9$

3. $y = -3x + 5$

   $4x + y = 8$

4. $x = y - 4$

   $2x - y = -5$

5. $y = 5x - 2$

   $-x + 2y = 10$

6. $x = -2y + 7$

   $3x + 4y = 15$

7. $y = 0.5x + 1$

   $x - 4y = -4$

🔑 Answers to Practice Quiz

1. $(2, 5)$
2. $(3, 2)$
3. $(3, -4)$
4. $(-1, 3)$
5. $(2, 8)$
6. $(1, 3)$
7. $(0, 1)$

💡 Tips & Tricks

• 🧐 Double-Check Your Work:
Careless errors can lead to incorrect solutions. Always verify your steps.
• 🧮 Simplify First:
If possible, simplify equations before substituting to make the algebra easier.
• 💪 Practice Makes Perfect:
The more you practice, the faster and more accurate you'll become.

🎓 Conclusion

Solving systems of equations by substitution, when a variable is already isolated, is a powerful tool in algebra. By understanding the key principles and practicing consistently, you can master this skill and apply it to various real-world problems. Keep practicing, and you'll be solving systems like a pro in no time!