Simultaneous Equations

Simultaneous Equations Overview

  • Simultaneous equations are a set of two or more equations, each containing two or more variables. The term ‘simultaneous’ indicates that the equations are to be solved together at the same time.
  • The solution of a system of simultaneous equations is an ordered pair (or triple for three variables) that satisfies all the equations in the system simultaneously.

Solving Simultaneous Equations

  • One common method to solve simultaneous equations is the substitution method, which involves solving one of the equations for one variable in terms of the other variables, and then substituting this expression into the other equations.
  • Another common method is the elimination method, where you eliminate one of the variables by adding or subtracting the equations.
  • For example, to solve the system of equations 2x + 3y = 12 and x - y = 2, you could add the two equations to get 3x + 2y = 14, from which x can be isolated and then substituted back into one of the original equations to find y.

Simultaneous Equations with Quadratics

  • A system of simultaneous equations may include a linear equation and a quadratic equation.
  • These are often solved using the substitution method. You solve the linear equation for one variable and substitute that expression into the quadratic equation.

Word Problems Involving Simultaneous Equations

  • Simultaneous equations are often used to solve word problems, with each equation representing a particular aspect of the problem.
  • For example, problems involving mixtures, rate, or cost could be solved using simultaneous equations.

Revision Tips and Practice

  • Always check your solutions by substituting them back into the original equations to make sure they satisfy both.
  • The only way to get better at solving simultaneous equations is consistent practice. The key is to recognise when to use the substitution method and when to use the elimination method.
  • Break down word problems into smaller parts if you’re finding them difficult. Create equations from each part, then combine them to form a system of simultaneous equations.

Always remember, simultaneous equations are a useful tool in algebra and are widely used to solve real-world problems across different aspects of life including engineering, business and physics.