Solving Linear Systems with the Elimination Method

In the realm of mathematics, linear systems are a fundamental concept that arises in various applications, from physics and engineering to economics and finance. A linear system is a set of equations, each representing a straight line, where the goal is to find the values of the unknown variables that satisfy all the equations simultaneously. One powerful technique for solving linear systems is the elimination method, which involves manipulating the equations to eliminate one variable at a time.

Understanding the Elimination Method

The elimination method is based on the principle of adding or subtracting equations in a way that eliminates one of the variables. To achieve this, we aim to create coefficients for one variable that are opposites in the two equations. This allows us to add the equations together, effectively canceling out the chosen variable.

Steps Involved in the Elimination Method

Follow these steps to solve a linear system using the elimination method:

1. Choose a variable to eliminate: Identify the variable that you want to eliminate. Look for equations where the coefficients of the chosen variable are either equal or opposites.
2. Multiply equations (if necessary): If the coefficients of the chosen variable are not equal or opposites, multiply one or both equations by a constant to make them so.
3. Add or subtract the equations: Add or subtract the equations together, depending on whether the coefficients are opposites or equal. This will eliminate the chosen variable.
4. Solve for the remaining variable: After eliminating one variable, you'll be left with a single equation with one unknown. Solve for this remaining variable.
5. Substitute to find the other variable: Substitute the value you just found back into one of the original equations to solve for the other variable.
6. Check your solution: Substitute the values you found for both variables back into all the original equations to verify that they satisfy the system.

Example: Solving a System of Two Equations

Let's consider the following system of equations:

Equation 1: 2x + 3y = 7

Equation 2: 4x - y = 1

Step 1: Choose a variable to eliminate

Let's choose to eliminate 'y'.

Step 2: Multiply equations (if necessary)

Multiply Equation 2 by 3 to make the coefficients of 'y' opposites:

Equation 2 (multiplied by 3): 12x - 3y = 3

Step 3: Add the equations

Add Equation 1 and the modified Equation 2:

(2x + 3y) + (12x - 3y) = 7 + 3

14x = 10

Step 4: Solve for the remaining variable

Divide both sides by 14:

x = 10/14 = 5/7

Step 5: Substitute to find the other variable

Substitute x = 5/7 into Equation 1:

2(5/7) + 3y = 7

10/7 + 3y = 7

3y = 7 - 10/7 = 39/7

y = 39/21 = 13/7

Step 6: Check your solution

Substitute x = 5/7 and y = 13/7 into both original equations to verify that they are satisfied.

Conclusion

The elimination method is a powerful and efficient technique for solving linear systems. By systematically eliminating variables, we can reduce the system to a single equation with one unknown, making it easy to solve. This method is particularly useful when dealing with systems of two or more equations. Understanding the steps and applying them with practice will enable you to confidently solve linear systems using the elimination method.