# Double Elimination: A Math Trick for Solving Systems of Equations

In the realm of algebra, solving systems of equations is a fundamental skill. These systems often represent real-world scenarios, making their solution crucial for understanding various applications. One powerful technique for tackling such systems is the double elimination method, a clever approach that leverages the power of algebraic manipulation.

## Understanding Systems of Equations

A system of equations is a set of two or more equations that share the same variables. The goal is to find the values of these variables that satisfy all the equations simultaneously. Let's consider a simple example:

Equation 1: 2x + y = 7

Equation 2: x - y = 2

Our objective is to determine the values of 'x' and 'y' that make both equations true.

## The Double Elimination Method

The double elimination method involves strategically manipulating the equations to eliminate one variable at a time. Here's a step-by-step guide:

**Step 1: Align the equations.**Ensure that the variables (x and y) are aligned in both equations. In our example, they are already aligned.**Step 2: Multiply equations to create opposite coefficients.**Our goal is to have the coefficients of one variable be opposites. In this case, the coefficients of 'y' are already opposites (1 and -1). If they weren't, we would multiply one or both equations by a suitable constant to achieve this.**Step 3: Add the equations.**Adding the equations together will eliminate the variable with opposite coefficients. In our example, adding the equations gives us:(2x + y) + (x - y) = 7 + 2

This simplifies to 3x = 9

**Step 4: Solve for the remaining variable.**We can now solve for 'x':3x = 9

x = 3

**Step 5: Substitute the value back into one of the original equations.**We can substitute 'x = 3' into either Equation 1 or Equation 2. Let's use Equation 1:2(3) + y = 7

6 + y = 7

y = 1

**Step 6: Verify the solution.**Substitute both 'x = 3' and 'y = 1' into both original equations to ensure they hold true.

## Example: Solving a More Complex System

Let's try a slightly more complicated system:

Equation 1: 3x + 2y = 11

Equation 2: 5x - 3y = 8

**Step 1: Align the equations.**The variables are already aligned.**Step 2: Multiply equations to create opposite coefficients.**To eliminate 'y', we can multiply Equation 1 by 3 and Equation 2 by 2:(3x + 2y) * 3 = 11 * 3

(5x - 3y) * 2 = 8 * 2

This gives us:

9x + 6y = 33

10x - 6y = 16

**Step 3: Add the equations.**Adding the equations eliminates 'y':(9x + 6y) + (10x - 6y) = 33 + 16

19x = 49

**Step 4: Solve for 'x'.**x = 49/19

**Step 5: Substitute the value of 'x' back into one of the original equations.**Using Equation 1:3(49/19) + 2y = 11

147/19 + 2y = 11

2y = 11 - 147/19

2y = 22/19

y = 11/19

**Step 6: Verify the solution.**Substitute 'x = 49/19' and 'y = 11/19' into both original equations to confirm they are satisfied.

## Benefits of Double Elimination

The double elimination method offers several advantages:

**Systematic approach:**It provides a clear and structured process for solving systems of equations.**Efficiency:**It can be more efficient than other methods, especially for complex systems.**Flexibility:**It can be adapted to solve systems with any number of variables.

## Conclusion

The double elimination method is a valuable tool in the algebra toolbox. By understanding its steps and applying it to practice problems, you can confidently solve systems of equations and unlock their applications in various fields.