Solving three linear simultaneous equations

Solving Three Linear Simultaneous Equations

Understanding Simultaneous Equations

  • Simultaneous equations are a set of equations with multiple unknowns that are solved together.
  • Each equation represents a line in you are dealing with 2 variables, a plane if you have 3 variables, and so on. The solution to the set of equations is the intersection of these elements.
  • In the case of three simultaneous equations, you’re finding a point of intersection for three planes.

Methods of Solving

  • Method of Substitution: One of the equations is solved for one of the variables, then that expression is substituted into the other equations to reduce the number of variables in those equations.
  • Method of Elimination: The equations are manipulated by adding, subtracting, or scaling, to eliminate one variable, reducing the number of variables in those equations.
  • Gaussian Elimination or Matrix Method: This uses a combination of scaling and substitution to solve for multiple variables efficiently by generating a matrix and then reducing it to Row-Echelon form.

Key Steps in the Process

  • Step 1: Look for opportunities to eliminate one variable by adding or subtracting the equations from each other.
  • Step 2: When one variable is eliminated from two of the equations, you will be left with a pair of simultaneous equations with two variables, which will be solvable by substitution or elimination.
  • Step 3: Solve the pair of simultaneous equations to find the value of two of the variables.
  • Step 4: Substitute these solutions into any of the original equations to find the value of the final variable.

Important Considerations

  • Always ensure the equations are in Standard Form (Ax + By + Cz = D) for ease of manipulation.
  • Check your solutions by substituting them back into the original equations to ensure they satisfy all three equations.
  • In some instances, there may be no solution or infinitely many solutions. If the planes do not intersect at a common point, there is no solution. If the three planes intersect in a line, there are infinitely many solutions.
  • Understanding the geometrical interpretation can often make identifying these exceptions easier.