Algebraic Solution to Simultaneous Equations (SQA National 5 Maths): Revision Note

Solving simultaneous equations algebraically

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 terms like x² or y² 

  • Simultaneous because they have to be solved at the same time, to find answers that work for both

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 signs in front of the terms 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 signs in front of the terms you want to eliminate are 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