Neither x term nor y term are the same

Introduction to Neither x term nor y term are the same

  • Situations observing neither x term nor y term being the same appear predominantly in simultaneous equations.
  • These equations are set such that the coefficients of x and y are not equal, making it difficult to use direct addition or subtraction for a solution.
  • Solving involves equating the coefficients of the x or y terms in both equations, which often requires multiplication or division.

Understanding the Principle

  • The goal is to manipulate the equations such that they can be added or subtracted, resulting in the elimination of one variable.
  • This requires making the coefficients of x or y in the two equations equal or opposite of each other.
  • For example, in the equations 2x + 3y = 10 and 3x - 2y = 18, neither the x term nor y term have the same coefficients.

Steps to Solve

  1. Identify the coefficients of the x term and the y term in both equations.
  2. Determine a common multiple for the coefficients of either x or y, or make one’s coefficient the negative of the other.
  3. Modify one or both equations by multiplying or dividing to ensure the coefficients of x or y are either the same or opposite.
  4. Add or subtract the equations to eliminate one variable.
  5. Solve for the remaining variable and substitute it back into one of the original equations to find the eliminated variable’s value.

Examples

  • For example, let’s consider the equations 2x + 3y = 20 and x - y = 5. These equations haven’t got similar coefficients for x or y.
  • Multiply the second equation by 2, it turns into 2x - 2y = 10. Now, we can subtract these from the first equation: (2x + 3y) - (2x - 2y) = 20 - 10, giving solution 5y = 10, or y = 2. Substitute y = 2 into the original equation x - y = 5 gives x = 7.

Key Things to Remember

  • Equating coefficients of variables and eliminating one variable is key in solving these kind of equations.
  • It is crucial to maintain the equations’ balance when performing operations like addition, subtraction, multiplication, or division.
  • Be cautious of the signs while adding or subtracting equations.