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:

1. Step 1: Align the equations. Ensure that the variables (x and y) are aligned in both equations. In our example, they are already aligned.
2. 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.
3. 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

4. Step 4: Solve for the remaining variable. We can now solve for 'x':

   3x = 9

   x = 3

5. 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

6. 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

1. Step 1: Align the equations. The variables are already aligned.
2. 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

3. Step 3: Add the equations. Adding the equations eliminates 'y':

   (9x + 6y) + (10x - 6y) = 33 + 16

   19x = 49

4. Step 4: Solve for 'x'.

   x = 49/19

5. 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

6. 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.