A2oz

How to Solve a System of Equations by Elimination?

Published in Algebra 2 mins read

Solving a system of equations by elimination involves manipulating the equations to eliminate one variable, allowing you to solve for the remaining variable. Here's a step-by-step guide:

1. Align the Equations

  • Arrange the equations vertically, aligning the terms with the same variables.
  • For example:
    • 2x + 3y = 7
    • x - y = 1

2. Multiply to Create Opposites

  • Multiply one or both equations by a constant to make the coefficients of one variable opposites.
  • In our example, multiply the second equation by 3:
    • 2x + 3y = 7
    • 3x - 3y = 3

3. Add the Equations

  • Add the corresponding terms of the two equations together. This eliminates the variable with opposite coefficients.
  • In our example, adding the equations gives:
    • 5x = 10

4. Solve for the Remaining Variable

  • Solve the resulting equation for the remaining variable.
  • In our example, divide both sides by 5 to get:
    • x = 2

5. Substitute to Find the Other Variable

  • Substitute the value you just found back into either of the original equations.
  • Using the first original equation, we get:
    • 2(2) + 3y = 7
    • 4 + 3y = 7
    • 3y = 3
    • y = 1

6. Write the Solution

  • The solution to the system of equations is the ordered pair (x, y). In our example, the solution is (2, 1).

Examples

Here are a few more examples to illustrate the process:

  • Example 1:
    • x + 2y = 5
    • 3x - y = 1
    • Multiply the second equation by 2:
      • x + 2y = 5
      • 6x - 2y = 2
    • Add the equations:
      • 7x = 7
    • Solve for x:
      • x = 1
    • Substitute x = 1 into the first equation:
      • 1 + 2y = 5
      • 2y = 4
      • y = 2
    • Solution: (1, 2)
  • Example 2:
    • 2x - 3y = 1
    • 4x + 5y = 13
    • Multiply the first equation by -2:
      • -4x + 6y = -2
      • 4x + 5y = 13
    • Add the equations:
      • 11y = 11
    • Solve for y:
      • y = 1
    • Substitute y = 1 into the first equation:
      • 2x - 3(1) = 1
      • 2x - 3 = 1
      • 2x = 4
      • x = 2
    • Solution: (2, 1)

Practical Insights

  • The elimination method is particularly useful when the coefficients of one variable are already opposites or can be easily made opposites.
  • If the coefficients are not opposites, you may need to multiply both equations by different constants to create opposites.
  • Always check your solution by substituting the values back into the original equations to verify their accuracy.

Related Articles