Linear Simultaneous Equations (Cambridge International IGCSE Further Maths)

Elimination Method

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

    • 3*x *+ 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?

Exam Hint

• Don't skip the checking step (it only takes a few seconds) – there are many places to go wrong when solving simultaneous equations!
• Mishandling minus signs is probably the single biggest cause of student error in simultaneous equations questions

Worked Example

Substitution Method

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

Exam Hint

• Although elimination will always work to solve simultaneous linear equations, sometimes substitution can be easier and quicker
• Knowing both methods can help you a lot in the exam (plus you will need substitution to solve quadratic simultaneous equations)

Worked Example