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IBMathsDPAnalysis & Approaches (AA)HLRevision Notes1. Number & AlgebraSystems of Linear EquationsSolving Systems Using Row Reduction

Solving Systems Using Row Reduction (DP IB Analysis & Approaches (AA)): Revision Note

Dan Finlay

Written by: Dan Finlay

Reviewed by: Mark Curtis

Updated on

DP Analysis & Approaches (AA): HLDP Analysis & Approaches (AA): HL

Row reduction

How do I write a system of linear equations as a matrix?

  • To save space you can just write the coefficients without the variables in a grid-like structure called a matrix

  • For 2 variables

    • a subscript 1 x plus b subscript 1 y equals c subscript 1 a subscript 2 x plus b subscript 2 y equals c subscript 2 can be written as open square brackets table row cell a subscript 1 end cell cell b subscript 1 end cell row cell a subscript 2 end cell cell b subscript 2 end cell end table stretchy vertical line space table row cell c subscript 1 end cell row cell c subscript 2 end cell end table close square brackets

  • For 3 variables

    • a subscript 1 x plus b subscript 1 y plus c subscript 1 z equals d subscript 1 a subscript 2 x plus b subscript 2 y plus c subscript 2 z equals d subscript 2 a subscript 3 x plus b subscript 3 y plus c subscript 3 z equals d subscript 3 can be written as open square brackets table row cell a subscript 1 end cell cell b subscript 1 end cell cell c subscript 1 end cell row cell a subscript 2 end cell cell b subscript 2 end cell cell c subscript 2 end cell row cell a subscript 3 end cell cell b subscript 3 end cell cell c subscript 3 end cell end table stretchy vertical line space table row cell d subscript 1 end cell row cell d subscript 2 end cell row cell d subscript 3 end cell end table close square brackets

What is a row-reduced system of linear equations?

  • A row-reduced system of linear equations has the form:

  • table attributes columnalign right center left columnspacing 0px end attributes row cell A subscript 1 x plus B subscript 1 y plus C subscript 1 z end cell equals cell D subscript 1 end cell row cell B subscript 2 y plus C subscript 2 z end cell equals cell D subscript 2 end cell row cell C subscript 3 z end cell equals cell D subscript 3 end cell end table

    • which corresponds to open square brackets table row cell A subscript 1 end cell cell B subscript 1 end cell cell C subscript 1 end cell row 0 cell B subscript 2 end cell cell C subscript 2 end cell row 0 0 cell C subscript 3 end cell end table stretchy vertical line space table row cell D subscript 1 end cell row cell D subscript 2 end cell row cell D subscript 3 end cell end table close square brackets

What row operations do I need to know?

  • Row operations are changes to rows that make the linear equations simpler to solve

    • They do not affect the solution

  • The rows are often labelled r subscript 1, r subscript 2 and r subscript 3 (or R subscript 1, R subscript 2 and R subscript 3)

  • You can switch any two rows

    • begin mathsize 14px style stretchy left square bracket table row cell a subscript 1 end cell cell b subscript 1 end cell cell c subscript 1 end cell row cell a subscript 2 end cell cell b subscript 2 end cell cell c subscript 2 end cell row cell a subscript 3 end cell cell b subscript 3 end cell cell c subscript 3 end cell end table stretchy vertical line space table row cell d subscript 1 end cell row cell d subscript 2 end cell row cell d subscript 3 end cell end table stretchy right square bracket end style  can be written as begin mathsize 14px style stretchy left square bracket table row cell a subscript 3 end cell cell b subscript 3 end cell cell c subscript 3 end cell row cell a subscript 2 end cell cell b subscript 2 end cell cell c subscript 2 end cell row cell a subscript 1 end cell cell b subscript 1 end cell cell c subscript 1 end cell end table stretchy vertical line space table row cell d subscript 3 end cell row cell d subscript 2 end cell row cell d subscript 1 end cell end table stretchy right square bracket end style using r subscript 1 left right arrow r subscript 3

      • This is useful for getting zeros to the bottom

      • Or getting a 1 to the top

  • You can multiply any row by a (non-zero) constant

    • begin mathsize 14px style stretchy left square bracket table row cell a subscript 1 end cell cell b subscript 1 end cell cell c subscript 1 end cell row cell a subscript 2 end cell cell b subscript 2 end cell cell c subscript 2 end cell row cell a subscript 3 end cell cell b subscript 3 end cell cell c subscript 3 end cell end table stretchy vertical line space table row cell d subscript 1 end cell row cell d subscript 2 end cell row cell d subscript 3 end cell end table stretchy right square bracket end style can be written as begin mathsize 14px style stretchy left square bracket table row cell a subscript 1 end cell cell b subscript 1 end cell cell c subscript 1 end cell row cell k a subscript 2 end cell cell k b subscript 2 end cell cell k c subscript 2 end cell row cell a subscript 3 end cell cell b subscript 3 end cell cell c subscript 3 end cell end table stretchy vertical line space table row cell d subscript 1 end cell row cell k d subscript 2 end cell row cell d subscript 3 end cell end table stretchy right square bracket end style using k r subscript 2 rightwards arrow r subscript 2

      • This is useful for getting a 1 as the first non-zero value in a row

  • You can add multiples of a row to another row

    • begin mathsize 14px style stretchy left square bracket table row cell a subscript 1 end cell cell b subscript 1 end cell cell c subscript 1 end cell row cell a subscript 2 end cell cell b subscript 2 end cell cell c subscript 2 end cell row cell a subscript 3 end cell cell b subscript 3 end cell cell c subscript 3 end cell end table stretchy vertical line space table row cell d subscript 1 end cell row cell d subscript 2 end cell row cell d subscript 3 end cell end table stretchy right square bracket end style can be written as begin mathsize 14px style stretchy left square bracket table row cell a subscript 1 end cell cell b subscript 1 end cell cell c subscript 1 end cell row cell a subscript 2 plus 5 a subscript 3 end cell cell b subscript 2 plus 5 b subscript 3 end cell cell c subscript 2 plus 5 c subscript 3 end cell row cell a subscript 3 end cell cell b subscript 3 end cell cell c subscript 3 end cell end table stretchy vertical line space table row cell d subscript 1 end cell row cell d subscript 2 plus 5 d subscript 3 end cell row cell d subscript 3 end cell end table stretchy right square bracket end style using r subscript 2 plus 5 r subscript 3 rightwards arrow r subscript 2

    • This is useful for creating zeros under a 1

How do I row reduce a system of linear equations?

  • STEP 1
    Get a 1 in the top left corner

    • begin mathsize 14px style stretchy left square bracket table row 1 cell B subscript 1 end cell cell C subscript 1 end cell row asterisk times asterisk times asterisk times row asterisk times asterisk times asterisk times end table stretchy vertical line space table row cell D subscript 1 end cell row asterisk times row asterisk times end table stretchy right square bracket end style

    • You can do this by dividing the row by the current value in its place

      • If the current value is 0 or an awkward number then you can swap rows first

  • STEP 2
    Get 0’s in the entries below the 1

    • You can do this by adding/subtracting a multiple of the first row to each row

      • begin mathsize 14px style stretchy left square bracket table row 1 cell B subscript 1 end cell cell C subscript 1 end cell row 0 asterisk times asterisk times row 0 asterisk times asterisk times end table stretchy vertical line space table row cell D subscript 1 end cell row asterisk times row asterisk times end table stretchy right square bracket end style

  • STEP 3
    Ignore the first row and column as they are now complete

    Repeat STEPS 1 - 2 to the remaining section

    • Get a 1: begin mathsize 14px style stretchy left square bracket table row 1 cell B subscript 1 end cell cell C subscript 1 end cell row 0 1 cell C subscript 2 end cell row 0 asterisk times asterisk times end table stretchy vertical line space table row cell D subscript 1 end cell row cell D subscript 2 end cell row asterisk times end table stretchy right square bracket end style

    • Then 0 underneath: begin mathsize 14px style stretchy left square bracket table row 1 cell B subscript 1 end cell cell C subscript 1 end cell row 0 1 cell C subscript 2 end cell row 0 0 asterisk times end table stretchy vertical line space table row cell D subscript 1 end cell row cell D subscript 2 end cell row asterisk times end table stretchy right square bracket end style

  • STEP 4
    Get a 1 in the third row

    • Using the same idea as STEP 1

      • begin mathsize 14px style stretchy left square bracket table row 1 cell B subscript 1 end cell cell C subscript 1 end cell row 0 1 cell C subscript 2 end cell row 0 0 1 end table stretchy vertical line space table row cell D subscript 1 end cell row cell D subscript 2 end cell row cell D subscript 3 end cell end table stretchy right square bracket end style

How do I solve a system of linear equations once it is in row-reduced form?

  • Once you row reduced the equations you can then convert it back to algebra

    • open square brackets table row 1 cell B subscript 1 end cell cell C subscript 1 end cell row 0 1 cell C subscript 2 end cell row 0 0 1 end table stretchy vertical line space table row cell D subscript 1 end cell row cell D subscript 2 end cell row cell D subscript 3 end cell end table close square brackets corresponds to table attributes columnalign right center left columnspacing 0px end attributes row cell x plus B subscript 1 y plus C subscript 1 z end cell equals cell D subscript 1 end cell row cell y plus C subscript 2 z end cell equals cell D subscript 2 end cell row z equals cell D subscript 3 end cell end table

  • Solve the equations starting at the bottom

    • You have the value for z from the third equation

    • Substitute z into the second equation and solve for y

    • Substitute z and y into the first equation and solve for x

Examiner Tips and Tricks

If one of the equations starts with a coefficient of 1 on the x, make that your first equation to save time!

Worked Example

Solve the following system of linear equations using algebra.

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x minus 3 y plus 4 z end cell equals 14 row cell x plus 2 y minus 2 z end cell equals cell negative 2 end cell row cell 3 x minus y minus 2 z end cell equals 10 end table

1-10-2-ib-aa-hl-row-reduction-we-solution

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    Author
    Reviewer
    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

    Mark Curtis

    Reviewer: Mark Curtis

    Expertise: Maths Content Creator

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.