Solve by elimination

Solve by Elimination: An Introduction

  • The method of elimination is often utilised when you have a system of linear equations.
  • The primary aim is to eliminate one variable by performing suitable operations - this could be either addition, subtraction, multiplication, or division.
  • Once one variable is eliminated, the equation reduces to a simple linear equation, which is easy to solve.

Solving Process for Elimination Method

  1. Given two equations, they can either have the same coefficients, or different.
  2. If the coefficients of one of the variables in both equations are the same, they can be subtracted to eliminate that variable.
  3. If the coefficients of one of the variables are not the same, one or both of the equations should be multiplied by suitable factors to make them the same. Then, proceed with subtraction or addition to eliminate that variable.
  4. Solve the reduced linear equation to find the value of the remaining variable.
  5. Substitute the found value into one of the original equations to find the value of the other variable.

Examples of the Elimination Method

  • Consider the system of equations 2x + 3y = 14 and 4x - 3y = 10. Here, the coefficients of y are opposites. Adding these equations eliminates y, leading to 6x = 24, hence, x = 4. Substituting x = 4 into the first equation gives 2(4) + 3y = 14, indicating that y = 2.

  • Consider the system 2x + 4y = 18 and 3x + 2y = 16. Here, the coefficients of y are not the same. Multiply the first equation by 2 and second by 4 to get the new system 4x + 8y = 36 and 12x + 8y = 64. Now, subtract the first from the second to get 8x = 28 and therefore, x = 3.5. Substitute x = 3.5 into the first original equation to get y = 3.

Key Points to Remember

  • The elimination method is typically used when coefficients of a variable in two equations are the same, opposites, or multiples of each other.
  • It is essential to accurately perform basic mathematical operations-subtraction, addition, multiplication, or division- while using elimination.
  • Always substitute the found value back into one of the original equations to validate your answer and find the value for the remaining variable.