Simultaneous Equations (Edexcel IGCSE Further Pure Maths)

Linear Simultaneous Equations

What are linear simultaneous equations?

• When there are two unknowns (say x and y) in a problem, we need two equations to be able to find them both: these are called simultaneous equations

  • you solve two equations to find two unknowns, x and y

    • for example, 3x + 2y = 11 and 2x - y = 5

    • the solutions are x = 3 and y = 1

• If they just have x and y in them (no x² or y² or xy etc) then they are linear simultaneous equations

How do I solve linear simultaneous equations by elimination?

• "Elimination" completely removes one of the variables, x or y

• To eliminate the x's from 3x + 2y = 11 and 2x - y = 5 

  • Multiply every term in the first equation by 2

    • 6x + 4y = 22

  • Multiply every term in the second equation by 3

    • 6x - 3y = 15

  • Subtract the second result from the first to eliminate the 6x's

    • leaving 4y - (-3y) = 22 - 15, i.e. 7y = 7

  • Solve to find y (y = 1) then substitute y = 1 back into either original equation to find x (x = 3)

• Alternatively, to eliminate the y's from 3x + 2y = 11 and 2x - y = 5 

  • Multiply every term in the second equation by 2

    • 4x - 2y = 10

  • Add this result to the first equation to eliminate the 2y's (as 2y + (-2y) = 0)

    • The process then continues as above

• Check that your final solutions satisfy both equations

How do I solve linear simultaneous equations by substitution?

• "Substitution" means substituting one equation into the other

• Solve 3x + 2y = 11 and 2x - y = 5 by substitution

  • Rearrange one of the equation into y = ... (or x = ...)

    • For example, the second equation becomes y = 2x - 5 

  • Substitute this into the first equation (replace all y's with 2x - 5 in brackets)

    • 3x + 2(2x - 5) = 11

  • Solve this equation to find x (x = 3), then substitute x = 3 into y = 2x - 5 to find y (y = 1)

• Check that your final solutions satisfy both equations

How do I use graphs to solve linear simultaneous equations?

• Plot both equations on the same set of axes

  • to do this, you can use a table of values or rearrange into y = mx + c if that helps

• Find where the lines intersect (cross over)

  • The x and y  solutions to the simultaneous equations are the x- and y-coordinates of the point of intersection

• e.g. to solve 2x - y = 3 and 3x + y = 7 simultaneously, first plot them both (see graph)

  • find the point of intersection, (2, 1)

  • the solution is x = 2 and y = 1

How do I solve linear simultaneous equations from worded contexts?

Quadratic Simultaneous Equations

What are quadratic simultaneous equations?

• When there are two unknowns (say x and y ) in a problem, we need two equations to be able to find them both: these are called simultaneous equations

• If there is an x² or y² or xy  in one of the equations then they are quadratic (or non-linear) simultaneous equations

How do I solve quadratic simultaneous equations?

• Use the method of substitution

  • Substitute the linear equation, y = ... (or x = ...), into the quadratic equation

    • Do not try to substitute the quadratic equation into the linear equation

• For example solve x² + y² = 25 and y - 2x = 5 

  • Rearrange the linear equation into y = 2x + 5

  • Substitute this into the quadratic equation, replacing all y's with (2x + 5) in brackets

    •  x² + (2x + 5)² = 25

  • Expand and solve this quadratic equation (x = 0 and x = -4)

  • Substitute each value of x into the linear equation, y = 2x + 5, to get their value of y

  • Present your solutions in a way that makes it obvious which x  belongs to which y

    • x = 0, y = 5 or x = -4, y = -3

• Check that your final solutions satisfy both equations

How do you use graphs to solve quadratic simultaneous equations?

• Plot both equations on the same set of axes

  • to do this, you can use a table of values (or, for straight lines, rearrange into y = mx + c  if it helps)

• Find where the lines intersect (cross over)

  • The x  and y  solutions to the simultaneous equations are the x- and y-coordinates of the point of intersection

• e.g. to solve y = x² + 3x + 1 and y = 2x + 1 simultaneously, first plot them both (see graph)
  

  • find the points of intersection, (-1, -1) and (0, 1)

  • the solutions are x = -1 and y = -1, or x = 0 and y = 1