Solving Systems of Equations by Elimination: Addition Method

📚 Introduction to Elimination (Addition Method)

The elimination method, also known as the addition method, is a technique used to solve systems of linear equations. It involves manipulating the equations so that when added together, one of the variables is eliminated, leaving a single equation with one variable. This makes it much easier to find the solution.

📜 History and Background

The concept of solving simultaneous equations has ancient roots, dating back to Babylonian mathematics. However, the systematic approach of elimination became more formalized in the 17th century with the development of algebra. Today, it’s a fundamental technique in linear algebra and used in various fields.

🔑 Key Principles of the Addition Method

• 🎯 Goal: Eliminate one variable by adding the equations together.
• ➕ Adding Equations: Ensure that the coefficients of one variable are opposites (e.g., 3x and -3x).
• ↔️ Multiplication: If necessary, multiply one or both equations by a constant to make the coefficients of one variable opposites.
• ✅ Verification: Substitute the solution back into the original equations to check for accuracy.

✍️ Step-by-Step Guide

1. 1️⃣ Align the Equations: Write the equations so that like terms are aligned in columns.
2. 2️⃣ Multiply (if needed): Multiply one or both equations by a constant so that the coefficients of one variable are opposites.
3. 3️⃣ Add the Equations: Add the equations together. This should eliminate one variable.
4. 4️⃣ Solve: Solve the resulting equation for the remaining variable.
5. 5️⃣ Substitute: Substitute the value back into one of the original equations to solve for the other variable.
6. 6️⃣ Check: Verify your solution by substituting both values into both original equations.

➗ Example 1: Simple Elimination

Solve the following system of equations:

$x + y = 5\newline x - y = 1$

1. Equations are already aligned.
2. Coefficients of $y$ are already opposites.
3. Add the equations: $(x + y) + (x - y) = 5 + 1$, which simplifies to $2x = 6$.
4. Solve for $x$: $x = 3$.
5. Substitute $x = 3$ into the first equation: $3 + y = 5$, so $y = 2$.
6. Solution: $x = 3$, $y = 2$.

✖️ Example 2: Requiring Multiplication

Solve the following system of equations:

$2x + y = 7\newline x + 3y = 7$

1. Multiply the second equation by -2: $-2(x + 3y) = -2(7)$, which gives $-2x - 6y = -14$.
2. Add the modified second equation to the first equation: $(2x + y) + (-2x - 6y) = 7 + (-14)$, which simplifies to $-5y = -7$.
3. Solve for $y$: $y = \frac{7}{5}$.
4. Substitute $y = \frac{7}{5}$ into the first equation: $2x + \frac{7}{5} = 7$, so $2x = 7 - \frac{7}{5} = \frac{28}{5}$, and $x = \frac{14}{5}$.
5. Solution: $x = \frac{14}{5}$, $y = \frac{7}{5}$.

🌍 Real-world Applications

• 💰 Economics: Modeling supply and demand curves.
• 🧪 Chemistry: Balancing chemical equations.
• ⚙️ Engineering: Solving for forces in static systems.

💡 Tips for Success

• 🔎 Check for Opposites: Always look for variables with opposite coefficients first.
• ✍️ Neatness Counts: Keep your work organized to avoid errors.
• 💯 Practice Makes Perfect: Solve plenty of problems to become proficient.

📝 Practice Quiz

Solve the following systems of equations using the elimination method:

1. $x + y = 8$, $x - y = 2$
2. $2x + 3y = 13$, $2x - y = 5$
3. $3x + 2y = 7$, $x + y = 3$
4. $4x - y = 10$, $2x + y = 2$
5. $5x + 2y = 12$, $x - y = -1$
6. $2x + 5y = 16$, $x + 3y = 10$
7. $3x - 4y = -6$, $x + 2y = 8$

✅ Conclusion

The elimination method is a powerful tool for solving systems of linear equations. With practice and a clear understanding of the steps, you can confidently tackle these problems. Keep practicing, and you'll master it in no time! 👍