Solving Linear Equations (Cambridge (CIE) IGCSE International 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

• Adding 1 was undone by subtracting 1

• Multiplying by 2 was undone by dividing by 2

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

  • Multiplication and division are also inverse operations

Does the order of steps matter?

• As long as each step is applied correctly, the order in which inverse operations are applied does not matter

  • Applying the operations in one order may be easier than another

• Consider

  • 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 

  • This is incorrect as 8 has not been divided by 4

  • You must divide every term by 4

• Then subtract 2 from both sides

How do I solve linear equations with negative numbers?

• For example,

  • Subtract 2 from both sides and simplify

• Then divide both sides by -3 and simplify

• Some people prefer to write as , which is equivalent

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

How do I solve linear equations with brackets?

• If a linear equation involves brackets, expand the brackets first

• For example, solve

  • Expand the brackets 

• Then solve as shown previously

  • Add 6 then divide by 2

• Expanding brackets first will always work, but you can also divide first

  • Dividing both sides of by 2 gives

    • which gives 

  • This method works but can lead to harder fractions

How do I solve linear equations with fractions?

• If a linear equation contains fractions, multiply both sides by the lowest common denominator

• For example,

  • The lowest common denominator of 5 and 2 is 10

  • Multiply all terms on both sides by 10

• Simplify the fractions

• Now solve as before, by subtracting 40, then dividing by 2

• Unless the question specifies otherwise, you can leave the answer like this

  • A decimal or mixed number would also be accepted

What if the unknown is on the denominator?

• For example 

  • Multiply both sides of the equation by the denominator

• Simplify the fractions, and expand any brackets

• Now solve as before, by adding 6 to both sides, then dividing by 3

How do I solve linear equations with x terms on both sides?

• Collect the x terms (or whichever variable is involved) together on one side 

  • To do this, remove all the x terms from one side

    • It is easiest to remove the smallest x term to avoid negatives

• For example,

• Remove the x term on the right-hand side, by subtracting x from both sides

• There are no longer any x terms on the right

• This now has the same form as previously seen

  • Solve by adding 7 then dividing by 3