Non-Linear Simultaneous Equations : Theory

• Non-Linear Simultaneous Equations are 2 equations, where one of the graphs at least one of the equations doesn't form a straight line
• A typical instance is when we have a linear equation and quadratic equation - The linear graph cross the quadratic graph in 2 place, so there will be 2 solutions to the simultaneous equations

  
• Another typical instance is when we have a linear equation and the equation of the circle - the linear graph crosses the circle in 2 places, so there will be 2 solutions to the simultaneous equations

  
• Its possible to only have 1 solution - for the quadratic graph and the graph of the circle, the lines are tangents, and they only touch the graphs in 1 place

  
• Its also possible to have no solutions - in the 2 graphs below, the linear graphs do not cross the other graphs

  
• If any of the following algebraic terms in are in an equation, you can be sure that the equation is non-linear

  
• • In order to solve non-linear equations, we have to eliminate one of the variables (x or y) 
  • This gives an equation, where only the other variable is missing, and we can rearrange it to find the value of the other variable
• • To solve simultaneous equations, written in the form y = x² + bx + c , and y = ex - d, as both equations are equal to y, they are equal to each other 
  • x² + bx + c = ex - d  - We can now rearrange the equation so that one side is 0 
  • x² + (b - e)x + (c + d) = 0 - We would then factorise the equation, to find the 2 values of x - this can sometimes only be 1 value 
  • Then we would substitute the 1 or 2 values of x into both equations, to find the values of y - if there is only one x value, substitute the value of x into only one equation

  
• • To find the coordinates of the points of intersection for the graphs of 2 simultaneous equations, you factorise the equations, to find 2 values of x 
  • You then substitute the x values into either of the equations, e.g. the linear one, to find the corresponding values of y 
  • You  then write the solutions in coordinate form

  
• • If you have y² in one of the simultaneous equations, and you are given the value of y, you can substitute this value into y² 
  • You can then find the values of x and y

  
• Here’s what to do when you have an x² and y² in one your simultaneous equations