Simultaneous Equations (Edexcel GCSE Maths: Foundation)

What are linear simultaneous equations?

• When there are two unknowns (x  and y ), we need two equations to find them both
  
  • For example, 3x  + 2y  = 11 and 2x  - y  = 5
    
    • The values that work are x  = 3 and y  = 1
• These are called linear simultaneous equations
  • Linear because there are no complicated terms like x²  or y² 

How do I solve linear simultaneous equations by elimination?

• Elimination removes one of the variables, x  or y
• To eliminate the x's from 3x  + 2y  = 11 and 2x  - y  = 5, make the number in front of the x (the coefficient) in both equations the same (the sign may be different)
  • Multiply every term in the first equation by 2
    • 6x  + 4y  = 22
  • Multiply every term in the second equation by 3
    • 6x  - 3y  = 15
  • Subtracting the second equation from the first eliminates x
    • When the sign in front of the term you want to eliminate is the same, subtract the equations

• • The y  terms have become 4y  - (-3y) = 7y  (be careful with negatives) 
    • Solve the resulting equation to find y
    • y  = 1
  • Then substitute y  = 1 into one of the original equations to find x
    • 3x  + 2 = 11 so 3x  = 9 giving x  = 3
  • Write out both solutions together, x  = 3 and y  = 1
• Alternatively, you could have eliminated the y's from 3x  + 2y  = 11 and 2x  - y  = 5 by making the coefficient of y  in both equations the same 
  • Multiply every term in the second equation by 2
  • Adding this to the first equation eliminates y  (and so on)
    • When the sign in front of the term you want to eliminate is different, add the equations

How do I solve linear simultaneous equations by substitution?

• Substitution means substituting one equation into the other
  • This is an alternative method to elimination
    
    • You can still use elimination if you prefer
• To solve 3x  + 2y  = 11 and 2x  - y  = 5 by substitution
  • Rearrange one of the equations into y  = ... (or x  = ...)
    • For example, the second equation becomes y  = 2x  - 5 
  • Substitute this into the first equation
  • This means 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

How do I solve linear simultaneous equations graphically?

• 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
• For example, to solve 2x  - y  = 3 and 3x  + y  = 7 simultaneously,
  • First plot them both on the same axes (see graph)
  • Find the point of intersection, (2, 1)
  • The solution is x  = 2 and y  = 1