Solving Linear Equations with the Elimination Method

Solving Linear Equations in Two Variables with the Elimination Method

When graphing or finding the solutions to systems of linear equations in two variables, the elimination method is a powerful technique that helps to simplify the problem. This strategy involves manipulating the equations to eliminate one variable, making it easier to solve for the other.

Setting up the equations

Suppose we have a system of two linear equations in two variables, (x) and (y), such as:

[ a_1x + b_1y = c_1 ]

[ a_2x + b_2y = c_2 ]

Choosing a strategy for elimination

To begin, we'll choose one of the equations as our base and identify which coefficient or constant term should be eliminated. The key here is to choose coefficients or constants that are not zero, so we can perform the necessary operations without divididing by zero.

Eliminating one variable

There are two primary approaches for eliminating one variable:

1. Eliminating by adding: We can add or subtract the two equations to make one coefficient of (x) or (y) zero, while keeping the other non-zero. Let's say we add (a_2) times the first equation to the second equation, such that:

[ (a_2a_1 + b_2)x + (a_2b_1)y = c_2 + a_2c_1 ]

Now we can solve for (y) (assuming (a_2b_1 \neq 0)) by dividing both sides by (a_2b_1):

[ y = \dfrac{1}{a_2b_1}(c_2 + a_2c_1 - a_2a_1x) ]

2. Eliminating by multiplying: We can multiply one of the equations by a constant factor to make one coefficient of (x) or (y) zero, while keeping the other non-zero. Let's say we multiply the first equation by (-b_2) and add it to the second equation:

[ (-b_2a_1 + b_1b_2)x + (b_2b_1)y = -b_2c_1 + b_1c_2 ]

Now we can solve for (y) (assuming (b_2b_1 \neq 0)) by dividing both sides by (b_2b_1):

[ y = \dfrac{1}{b_2b_1}(b_1c_2 - b_2c_1 - b_2a_1x) ]

Solving for the remaining variable

After eliminating one variable, we'll have an equation involving only one of the variables, which we can solve for that variable. Then we can substitute this value back into one of the original equations to find the other variable.

Special cases

The elimination method works best when the system has a unique solution. However, it can also be used to identify special cases such as:

1. Inconsistent system: The system has no solutions, and the elimination method will result in an equation that is always false.
2. Dependent system: The system has infinitely many solutions, and the elimination method will result in an equation that is always true.
3. Identical equations: The system represents the same line, and the elimination method will result in an equation that is always true or false, depending on the initial strategy.

Conclusion

The elimination method is a powerful tool for solving systems of linear equations in two variables. While there are other techniques, such as substitution and elimination by graphing, elimination is a straightforward and efficient method that can be applied to a wide variety of systems. With practice, you'll find that the elimination method becomes a quick and reliable strategy for solving these types of equations.