Simultaneous equations (Higher Tier)

Understanding Simultaneous Equations (Higher Tier)

  • Simultaneous equations are a set of equations with multiple variables that are all satisfied by the same values of the variables.
  • To solve simultaneous equations, you need to find the values of the variables that make all the equations in the set true at the same time.
  • Simultaneous equations can be solved graphically or algebraically. The graphical method involves drawing the lines or curves from each equation on a graph and finding the points of intersection, which represent the solutions.
  • The algebraic methods to solve simultaneous equations are the elimination method, substitution method, and matrix method.

Solving Simultaneous Equations Using the Elimination Method

  • The elimination method is often used when the coefficients of one of the variables are the same or easily made the same through multiplication or division.
  • To use the elimination method, add or subtract the equations to eliminate one of the variables, then solve for the remaining variable.
  • After finding the value of one variable, substitute this value into one of the original equations to find the value of the other variable.

Solving Simultaneous Equations Using the Substitution Method

  • The substitution method is suitable when one of the equations can be easily rearranged to express one variable in terms of the other.
  • Start by expressing one variable in terms of the other variable in one equation. Then, substitute this into the other equation to form an equation with one variable and solve for this variable.
  • Once the value of one variable is found, substitute this value into either original equation to determine the value of the other variable.

Solving Simultaneous Equations Using the Matrix Method

  • The matrix method involves representing the system of simultaneous equations as a matrix, then solving the matrix to find the values of the variables.
  • This method is particularly suitable for systems with more than two equations or two variables.
  • Be cautious with this method, as it requires a good understanding of matrix operations.

Solving Simultaneous Equations with Non-linear Terms

  • Simultaneous equations may contain quadratic or other non-linear terms. This creates a system of non-linear simultaneous equations.
  • Solving non-linear simultaneous equations usually requires a combination of algebraic and graphical methods.
  • When graphically representing these equations, the solutions are the points of intersection between the curves or lines represented by the equations.
  • Algebraically, one technique is to rearrange one of the equations to express one variable in terms of the other, then substitute this into the other equation similar to the substitution method.