Linear Simultaneous Equations (Cambridge O Level Additional Maths)

What are simultaneous linear 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
• Solving a pair of simultaneous equations means finding pairs of values that make both equations true at the same time
• A linear equation in two unknowns will produce a straight line if you graph it... linear = line
• A pair of simultaneous equations will produce lines that will cross each other (if there is a solution!)

How do I use elimination to solve simultaneous linear equations?

• "Elimination" completely removes one of the variables, x or y
• Begin by multiplying one (or both) of the equations by a constant (or constants)to get the numbers in front of one of the unknowns to match
  • For example 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
• If the matching numbers have the same sign, then subtract one equation from the other
• If the matching numbers have different signs then add the equations together
  • Subtract the second result from the first to eliminate the 6x's
    • 4y - (-3y) = 22 - 15
    • 7y = 7
• Solve the new equation to find the value of one of the unknowns
  • Solve to find y
    • y = 1
• Substitute the value into one of the original equations and solve to find the value of the other unknown
  • Substitute y = 1 back into either original equation
    • 3x + 2(1) = 11
    • 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
  • 3(3) + 2(1) = 11 and 2(3) - (1) = 5

How do I solve linear simultaneous equations from worded contexts?

How do I use substitution to solve simultaneous linear equations?

• "Substitution" means substituting one equation into the other
• Rearrange one of the equations to make one of the unknowns the subject 
  • To 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 the expression found for x or y into the equation not used to rearrange
  • Replace all y's with 2x - 5 in brackets
    • 3x + 2(2x - 5) = 11
• Solve the new equation to find the value of one of the unknowns
  • Solve this equation to find x
    • x = 3
• Substitute the value found for x or y into the rearranged equation from to find the value of the other unknown
  • Substitute x = 3 into y = 2x - 5 to find y
    • y = 2(3) - 5
    • y = 1
• Check your final solutions satisfy both equations
  • • 3(3) + 2(1) = 11 and 2(3) - (1) = 5

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 = 4 simultaneously, first plot them both (see graph)
  
  • find the point of intersection, (2, 1)
  • the solution is x = 2 and y = 1