What is the elimination method for systems (Grade 8, multiplying equations)?

📚 What is the Elimination Method?

The elimination method is a way to solve systems of equations by adding or subtracting the equations to eliminate one of the variables. This makes it easier to solve for the remaining variable.

📜 History and Background

The concept of solving systems of equations has been around for centuries. Ancient civilizations used similar methods to solve practical problems involving multiple unknowns. The formalization of the elimination method, as we know it today, evolved with the development of algebra.

💡 Key Principles

• 🎯 Goal: Eliminate one variable by manipulating the equations.
• 🔢 Multiply: Multiply one or both equations by a constant so that the coefficients of one variable are opposites.
• ➕ Add: Add the equations together. One variable should cancel out.
• ✏️ Solve: Solve for the remaining variable.
• ↩️ Substitute: Substitute the value back into one of the original equations to find the value of the eliminated variable.

🧮 Real-World Examples

Example 1:

Solve the system of equations:

$2x + y = 7$

$x - y = 2$

1. The coefficients of $y$ are already opposites, so add the equations:

$(2x + y) + (x - y) = 7 + 2$

$3x = 9$

$x = 3$

2. Substitute $x = 3$ into the second equation:

$3 - y = 2$

$y = 1$

Solution: $x = 3$, $y = 1$

Example 2: Multiplying Equations

Solve the system of equations:

$x + 2y = 5$

$3x - y = 1$

1. Multiply the second equation by 2 to make the coefficients of $y$ opposites:

$2(3x - y) = 2(1)$

$6x - 2y = 2$

2. Add the modified second equation to the first equation:

$(x + 2y) + (6x - 2y) = 5 + 2$

$7x = 7$

$x = 1$

3. Substitute $x = 1$ into the first equation:

$1 + 2y = 5$

$2y = 4$

$y = 2$

Solution: $x = 1$, $y = 2$

✍️ Practice Problems

Solve the following systems of equations using the elimination method:

1. $3x + 2y = 8$ and $x - 2y = 0$
2. $2x + y = 7$ and $x + 3y = 7$
3. $4x - 3y = 10$ and $x + y = 1$

✅ Conclusion

The elimination method is a powerful tool for solving systems of equations. By strategically multiplying and adding equations, you can simplify complex problems and find solutions efficiently. Remember to check your answers by substituting them back into the original equations.