๐ Understanding the Elimination Method
The elimination method, also known as the addition method, is a technique used to solve systems of linear equations. The core idea is to manipulate the equations in such a way that when you add them together, one of the variables is eliminated, leaving you with a single equation in one variable that can be easily solved. This method is particularly useful when the coefficients of one of the variables are opposites or can easily be made opposites.
๐ A Brief History
While the concept of solving simultaneous equations dates back to ancient civilizations, the formalization of methods like elimination developed over centuries. Mathematicians like Carl Friedrich Gauss contributed significantly to the development of linear algebra and related techniques in the 18th and 19th centuries, solidifying the elimination method as a fundamental tool.
๐ Key Principles of Elimination
- ๐ฏ Identify the Target Variable: Decide which variable you want to eliminate. Look for variables with coefficients that are the same or easily made the same (but with opposite signs).
- โ๏ธ Manipulate Equations: Multiply one or both equations by a constant so that the coefficients of the target variable are opposites. For example, if you have $2x + y = 5$ and $x - y = 1$, you can directly add the equations to eliminate $y$. If you have $2x + 3y = 7$ and $x + y = 3$, multiply the second equation by -2 to get $-2x - 2y = -6$.
- โ Add the Equations: Add the modified equations together. The target variable should cancel out.
- ๐ Solve for the Remaining Variable: Solve the resulting equation for the remaining variable.
- ๐ Substitute and Solve: Substitute the value you found back into one of the original equations to solve for the other variable.
- โ๏ธ Check Your Solution: Plug both values into both original equations to ensure they hold true.
โ Real-World Examples
Example 1: Buying Fruits
Suppose you buy 2 apples and 3 bananas for $5.50, and your friend buys 3 apples and 1 banana for $4.50. Let $a$ be the price of an apple and $b$ be the price of a banana. We can set up the following system of equations:
- ๐ Equation 1: $2a + 3b = 5.50$
- ๐ Equation 2: $3a + b = 4.50$
To eliminate $b$, multiply Equation 2 by -3:
- ๐ Equation 1: $2a + 3b = 5.50$
- โ Modified Equation 2: $-9a - 3b = -13.50$
Add the equations:
$-7a = -8$
$a = \frac{-8}{-7} \approx 1.14$
Substitute $a$ back into Equation 2:
$3(1.14) + b = 4.50$
$3.42 + b = 4.50$
$b = 1.08$
Therefore, an apple costs approximately $1.14 and a banana costs $1.08.
Example 2: Solving a System of Equations Directly
Solve the following system:
- ๐ข Equation 1: $x + y = 10$
- โ Equation 2: $x - y = 4$
Here, the $y$ terms are already opposites, so we can simply add the equations:
$2x = 14$
$x = 7$
Substitute $x$ back into Equation 1:
$7 + y = 10$
$y = 3$
Solution: $x = 7$, $y = 3$.
โ๏ธ Conclusion
The elimination method provides a powerful and systematic approach to solving systems of equations. By strategically manipulating and combining equations, you can efficiently find solutions to various problems in mathematics and real-world applications. Practice is key to mastering this valuable technique!