Solving Linear Systems: Elimination Method

In the realm of algebra, solving systems of linear equations is a fundamental skill. One powerful technique for tackling these systems is the elimination method. This method involves manipulating the equations to eliminate one variable, allowing you to solve for the remaining variable. Once you've found the value of one variable, you can substitute it back into either of the original equations to find the value of the other variable. Let's delve into the details of this method with illustrative examples.

Understanding the Elimination Method

The elimination method hinges on the principle of adding or subtracting equations in a way that eliminates one of the variables. To achieve this, you might need to multiply one or both equations by a constant. The goal is to create coefficients for one of the variables that are opposites, so when you add the equations together, that variable disappears.

Step-by-Step Guide

1. Identify the Variable to Eliminate: Look for a pair of variables with coefficients that are either the same or opposites. If necessary, multiply one or both equations by a constant to create coefficients that are opposites.
2. Add or Subtract the Equations: Add or subtract the equations together, depending on whether the coefficients are opposites or the same. The goal is to eliminate one of the variables.
3. Solve for the Remaining Variable: After eliminating one variable, you'll have a single equation with one unknown. Solve this equation for the remaining variable.
4. Substitute to Find the Other Variable: Substitute the value you just found back into either of the original equations. Solve for the remaining variable.
5. Check Your Solution: Substitute both values you found back into both original equations. If both equations are true, your solution is correct.

Illustrative Example

Let's consider the following system of equations:

Equation 1: 2x + 3y = 7

Equation 2: 4x - 3y = 1

Step 1: Notice that the coefficients of 'y' are opposites (3 and -3). We can proceed directly to step 2.

Step 2: Add the equations together:

(2x + 3y) + (4x - 3y) = 7 + 1

This simplifies to 6x = 8

Step 3: Solve for 'x':

x = 8/6 = 4/3

Step 4: Substitute the value of 'x' (4/3) back into either of the original equations. Let's use Equation 1:

2(4/3) + 3y = 7

8/3 + 3y = 7

3y = 13/3

y = 13/9

Step 5: Check the solution by substituting both values (x = 4/3, y = 13/9) into both original equations. You'll find that both equations hold true, confirming our solution.

Conclusion

The elimination method offers a systematic approach to solving systems of linear equations. By strategically manipulating the equations to eliminate one variable, we can solve for the remaining variable and subsequently find the complete solution. This method is a valuable tool in algebra and finds applications in various fields, including economics, engineering, and physics.