Simultaneous Equations (AQA GCSE Maths)

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 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 your final solutions satisfy both equations

How do you 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?

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
• 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 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