Solving Linear Equations (Edexcel GCSE Maths)

Solving Linear Equations

What are linear equations?

• A linear equation is one that can be written in the form

  • and  are numbers and is the variable

    • 2x + 3 = 5

    • 3x + 4 = 1

    • x - 5 = -3

• The greatest power of x is 1

  • There are no terms like x²

How do I solve linear equations?

• You need to use operations like adding, subtracting, multiplying and dividing to get x on its own

• Any operation you do to one side of the equation must also be done to the other side

• For example, to solve  look at the +1 on the left

  • Undo this by subtracting 1 from both sides and simplifying

• This equation is now easier to solve

• 2x is 2 × x  so undo this by dividing both sides by 2 and simplifying

• The solution to the equation is x = 4

• Note that +1 was undone by subtracting 1

  • addition and subtraction are said to be inverse (opposite) operations

• Similarly, multiplying by 2 was undone by dividing by 2

  • multiplication and division are also inverse operations

Does the order of operations matter?

• Take 

  • It is easier to first subtract 8 from both sides

• then divide both sides by 4

• If you want to first divide by 4, a common mistake is to write 

  • You cannot divide just the 4x and the 12 by 4

    • You must also divide the 8 by 4

• Then subtract 2 from both sides

How do I solve linear equations with negative numbers?

• For example, solve the equation

  • Subtract 2 from both sides and simplify

• Be careful not to lose the negative sign on the 3

• Then divide both sides by -3 and simplify

• An alternative way to think about  is by reordering the left-hand side

  • is the same as 

  • Reordering like this does not change the right-hand side

    • So the equation is

  • Some people prefer to see it like this

    • You then subtract 2 and divide by -3 as before