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Solving Linear Systems: Elimination Method

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Solving Linear Systems: Elimination Method

In algebra, a system of linear equations is a collection of two or more linear equations that share the same variables. Solving a system of linear equations means finding the values of the variables that satisfy all the equations simultaneously. There are several methods for solving linear systems, one of which is the elimination method.

Understanding the Elimination Method

The elimination method works by manipulating the equations in a system to eliminate one of the variables, allowing you to solve for the other. This is achieved by adding or subtracting the equations together in a way that cancels out one of the variables.

Steps for Solving Linear Systems by Elimination

Here's a step-by-step guide to solving linear systems using the elimination method:

  1. Align the Equations: Write the equations so that the variables are aligned vertically. For example:
2x + 3y = 7
4x - y = 1
  1. Multiply to Create Opposites: If the coefficients of one variable in the two equations are not opposites, multiply one or both equations by a constant to make them opposites. For example, to eliminate 'y' in the example above, multiply the second equation by 3:
2x + 3y = 7
12x - 3y = 3
  1. Add or Subtract Equations: Add the two equations together. The goal is to eliminate one of the variables. In our example, adding the equations eliminates 'y':
14x = 10
  1. Solve for the Remaining Variable: Solve the resulting equation for the remaining variable. In our example, divide both sides by 14 to find 'x':
x = 10/14 = 5/7
  1. Substitute to Find the Other Variable: Substitute the value you found for one variable back into either of the original equations to solve for the other variable. Let's substitute 'x = 5/7' into the first original equation:
2(5/7) + 3y = 7
10/7 + 3y = 7
3y = 39/7
y = 13/7
  1. Write the Solution: The solution to the system of equations is the ordered pair (x, y). In our example, the solution is (5/7, 13/7).

Example

Solve the following system of equations using the elimination method:

3x + 2y = 11
2x - 3y = -4

Step 1: The equations are already aligned.

Step 2: Multiply the first equation by 3 and the second equation by 2 to make the coefficients of 'y' opposites:

9x + 6y = 33
4x - 6y = -8

Step 3: Add the equations together to eliminate 'y':

13x = 25

Step 4: Solve for 'x':

x = 25/13

Step 5: Substitute 'x = 25/13' back into the first original equation:

3(25/13) + 2y = 11
75/13 + 2y = 11
2y = 58/13
y = 29/13

Step 6: The solution to the system is (25/13, 29/13).

Summary

The elimination method is a powerful technique for solving systems of linear equations. It involves manipulating the equations to eliminate one variable, allowing you to solve for the remaining variable. This method is particularly useful when the coefficients of one variable are already opposites or can easily be made opposites through multiplication.

Elimination Method