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Solving Linear Equations (AQA GCSE Maths)

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Amber

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Amber

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Solving Linear Equations

What are linear equations?

  • A linear equation is an equation that will produce a straight line when plotted on a graph
  • The greatest power of x in a linear equation is 1
    • This means there are no terms of space x squared or a higher order
  • A linear equation is normally in form a x space plus space b space equals space c
    • where a comma space b comma and c are constants and x spaceis a variable

How do I solve a linear equation?

  • To solve a linear equation you need to isolate the variable, usually x, by carrying out inverse operations to both sides of the equation
    • Inverse operations are just the opposite operations to what has already happened to the variable
  • The order in which the inverse operations are carried out is important
    • Most of the time, this will be BIDMAS in reverse
    • However it depends on the order in which the operations were applied to the variable to form the equation

How do I solve a linear equation of the form ax + b = c?

  • The operations that have been applied to x here are:
    • Multiply by a then add b
  • To solve this, you must carry out the inverse operations in reverse order
    • Subtract b then divide by a
    • The order in which you perform the inverse operations is important
  • For example, to solve the equation 2 x space plus space 1 space equals space 9
    • Subtract 1

table row cell 2 x space plus space 1 end cell equals cell space 9 space space space space space end cell end table
table row blank blank cell left parenthesis space minus space 1 right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space end cell end table space space table row blank blank cell space space space space space space space space space space space space space space space space space space space left parenthesis space minus 1 right parenthesis end cell end table table row blank blank end table
table row cell 2 x space end cell equals cell space 8 space end cell end table

    • Divide by 2

table row cell 2 x space end cell equals cell space 8 space space space space space end cell end table
table row blank blank cell left parenthesis divided by 2 right parenthesis space space space space space space space space space space end cell end table space space table row blank blank cell space space space space space space space space space space space space space left parenthesis divided by 2 right parenthesis end cell end table
table row cell x space end cell equals cell space 4 end cell end table

  • Be extra careful if any of the terms have negatives
  • For example, to solve the equation 2 space minus space 3 x space equals space 10
    • Subtract 2

table row cell 2 space minus space 3 x space end cell equals cell space 10 end cell end table
table row blank blank cell left parenthesis space minus 2 right parenthesis space space space space space space space space space space space space space space space space space space space space space end cell end table space space table row blank blank cell space space space space space space space space space space space space space space space space space space space space space left parenthesis space minus 2 right parenthesis end cell end table
table row cell negative 3 x space end cell equals cell space 8 end cell end table

      • Be careful not to drop the negative sign on -3
    • Divide by -3

table row cell negative 3 x space end cell equals cell space 8 end cell end table
table row blank blank cell space left parenthesis divided by negative 3 right parenthesis space space space space space space space space space space space space space space space space space space space end cell end table space space table row blank blank cell space space space space space space space space space space space space space space space space space space space space left parenthesis divided by negative 3 right parenthesis end cell end table
table row cell x space end cell equals cell negative 8 over 3 end cell end table

How do I solve a linear equation with the unknown variable, x, on both sides?

  • If a linear equation contains the unknown variable, usually x, on both sides start by collecting these terms together on one side of the equation
    • Moving the x term with the smallest coefficient (number in front of x) is easiest 
  • For example, to solve the equation  4 x space minus space 7 space equals space 11 space plus space x
    • STEP 1
      Move the x term with the smallest coefficient of x
      • The coefficients are 4 and 1 so move the x-term on the right hand side

table row cell space 4 x space minus space 7 space end cell equals cell space 11 space plus space x end cell end table
table row blank blank cell open parentheses negative x close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open parentheses negative x close parentheses space end cell end table
table row cell 3 x space minus space 7 space end cell equals cell space 11 end cell end table

    • STEP 2
      Solve the linear equation using the method above         

table row cell 3 x space minus space 7 space end cell equals cell space 11 end cell row cell 3 x space end cell equals cell space 18 end cell row cell x space end cell equals cell space 6 end cell end table

How do I solve a linear equation that contains brackets?

  • If a linear equation contains brackets on one, or both, sides start by expanding the brackets
  • For example, to solve the equation 2 open parentheses x space minus space 1 close parentheses space equals space 10 space minus open parentheses 3 space plus space x close parentheses
    • STEP 1
      Expand the brackets on both sides 

space space space space space space space space space space space space space space space space space space space 2 x space minus space 2 space equals space 10 space minus 3 space minus space x

    • STEP 2
      Collect the xterms to one side by subtracting the term with the smaller coefficient of x

table row cell 2 x space minus space 2 space end cell equals cell space 7 space minus space x end cell end table
table row blank blank cell negative space open parentheses negative x close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space minus space open parentheses negative x close parentheses end cell end table
table row cell 3 x space minus space 2 space end cell equals cell space 7 end cell end table

      • Be extra careful if any of the terms have negatives
    • STEP 3
      Solve the linear equation using the method above 

        table attributes columnalign right center left columnspacing 0px end attributes row cell 3 x space minus space 2 space end cell equals cell space 7 end cell row cell 3 x space end cell equals cell space 9 end cell row cell x space end cell equals cell space 3 end cell end table

How do I solve a linear equation that contains fractions?

  • If a linear equation contains a fraction on one or both sides, remove the fractions by multiplying both sides by everything on the denominator
    • Remember to put brackets around the expression first  
  • For example, to solve the equation fraction numerator 2 over denominator x space plus space 3 end fraction space equals space fraction numerator space 3 over denominator 4 space minus space 2 x end fraction you will need to multiply both sides of the equation by both open parentheses x space plus space 3 close parentheses and open parentheses 4 space minus space 2 x close parentheses 
    • STEP 1
      Multiply both sides by open parentheses x space plus space 3 close parentheses

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 2 over denominator x space plus space 3 end fraction open parentheses x space plus space 3 close parentheses space end cell equals cell space fraction numerator space 3 over denominator 4 space minus space 2 x end fraction open parentheses x space plus space 3 close parentheses end cell row cell 2 space end cell equals cell fraction numerator space 3 open parentheses x space plus space 3 close parentheses over denominator 4 space minus space 2 x end fraction end cell end table

    • STEP 2
      Multiply both sides by open parentheses 4 space minus space 2 x close parentheses

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 open parentheses 4 space minus space 2 x close parentheses end cell equals cell fraction numerator space 3 open parentheses x space plus space 3 close parentheses over denominator 4 space minus space 2 x end fraction open parentheses 4 space minus space 2 x close parentheses end cell row cell 2 open parentheses 4 space minus space 2 x close parentheses space end cell equals cell space 3 open parentheses x space plus space 3 close parentheses end cell end table

    • STEP 3
      Expand the brackets on both sides 

space space space space space space space space space space space space space space space space space space space 8 space minus space 4 x space equals space 3 x space plus space 9

    • STEP 4
      Collect the xterms to one side by subtracting the term with the smaller coefficient of x

table row cell 8 space minus space 4 x space end cell equals cell space 3 x space plus space 9 end cell end table
table row blank blank cell negative space open parentheses negative 4 x close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space minus space open parentheses negative 4 x close parentheses end cell end table
table row cell 8 space end cell equals cell space 7 x space plus space 9 end cell end table

    • STEP 5
      Solve the equation
      • You can swap the sides if it makes solving the equation easier    

    table row cell 7 x space plus space 9 space end cell equals cell space 8 end cell row cell 7 x space end cell equals cell space minus 1 end cell row cell x space end cell equals cell space minus 1 over 7 end cell end table

Examiner Tip

  • If you have time in the exam, you should substitute your answer back into the equation to check you got it right

Worked example

(a)
Solve the equation.

fraction numerator 3 x space minus space 2 over denominator 4 space minus space x end fraction space equals space minus 1

Get rid of the fraction by multiplying both sides by the denominator open parentheses 4 space minus space x close parentheses.

table row cell 3 x space minus space 2 space space end cell equals cell space minus open parentheses 4 space minus space x close parentheses end cell end table

Expand the brackets.

table row cell 3 x space minus space 2 space space end cell equals cell space minus 4 space plus space x end cell end table
 

Bring the x terms to one side of the equation by subtracting x from both sides.
 

table row cell 2 x space minus space 2 space space end cell equals cell space minus 4 space end cell end table

Get the x term by itself by adding 2 to both sides. 
 

2 x space equals space minus 2

Solve by dividing both sides by 2.

bold italic x bold equals bold minus bold 1

(b)
Solve the equation.

2 space plus space fraction numerator x space plus space 1 over denominator 3 end fraction space equals space 4

Isolate the fraction by subtracting 2 from both sides.

table row cell fraction numerator x space plus space 1 over denominator 3 end fraction space end cell equals cell space 2 end cell end table

Get rid of the fraction by multiplying both sides by the denominator (3).

table row cell x space plus space 1 space end cell equals cell space 6 end cell end table
 

Get the x term by itself by subtracting 1 from both sides. 
 

bold italic x bold space bold equals bold space bold 5

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    Amber

    Author: Amber

    Expertise: Maths

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.