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 we solve linear simultaneous equations by elimination method?

• “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 we solve linear simultaneous equations by the substitution method?

• “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 we solve linear simultaneous equations by graphical method?

• Plot both equations on the same set of axes
  • to do this, you can use a table of values or rearrange it 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 = 4 simultaneously, first plot them both (see graph)
  • find the point of intersection, (2, 1)
  • the solution is x = 2 and y =