Substitution methods (Quadratic SImultaneous)

Substitution methods (Quadratic SImultaneous)

Substitution Methods in Quadratic Simultaneous Equations

Overview

  • Quadratic simultaneous equations contain one linear and one quadratic equation.
  • The process of solving them often involves using a method called substitution.

The Substitution Method

  • Substitution is a method where you solve one equation for one variable in terms of the other variable.
  • This method is primarily used when it is easy to express one variable in terms of the other.

Using the Substitution Method

  1. Identify the linear and the quadratic equation.
  2. Solve the linear equation for one variable in terms of the other.
  3. Substitute this expression into the quadratic equation.
  4. This will result in a quadratic equation in one variable.
  5. Solve the quadratic equation using either factorisation, completing the square or the quadratic formula.
  6. Once the value of one variable is found, substitute it into one of the original equations to find the value of the other variable.

Example of Substitution Method

  • For the set of equations: y = 2x + 3 and y^2 = 4x + 12
  • The linear equation y = 2x + 3 can be solved to give x = (y - 3) / 2.
  • Substituting this into the quadratic equation gives ((y - 3) / 2)^2 = 4((y - 3) / 2) + 12.
  • This can be solved to find the values of y.
  • Substituting the y-values back into y = 2x + 3 will give the corresponding x-values.

Key Points to Remember

  • Be careful with your arithmetic and algebra when executing the substitution method, as small mistakes can lead to incorrect solutions.
  • Always check your solutions by substitifying them back into the original equations.
  • It’s not always possible to find real solutions to quadratic simultaneous equations, sometimes the solutions to these equations can be complex numbers.

Conclusion

  • Understanding the process of the substitution method in quadratic simultaneous equations is vital in algebra. The more practice you get, the more comfortable you’ll be with these kinds of problems. It enhances the critical thinking and problem-solving skills, which are necessary for advanced mathematical subjects.