[S2_20150520SERC] Simultaneous Equations with Reciprocals

Question

Introduction

     This is a secondary 2 (» grade 8) problem that involves linear simultaneous equations in two unknown.  It looks non-linear and indeed if you try to solve for  x  and  y  directly,  you might get into a big mess with extraneous solutions to boot.  What should our approach be?

Observations

     It is always a good idea to stare at the question for a little while longer before jumping in to try to solve it.  Let us (mentally) reformat the equations a little bit.

     Do you notice any repeated chunks?  Chunking is very useful.  Let use substitutions for those chunks to simplify the equations.

Solution

Final Remarks

     To recap: It is a good learn to make observations first before attempt to solve the problem.  If you see any repeated chunks, it is a good idea to use substitution to simplify the problem.  Once the reciprocals have been substituted, we try to use elimination (which is usually the more effective method).  Also, try as far as possible to avoid fractions.  By multiplying equation [2] by 5,  we get – 30Y, which can be cancelled if we multiply  15Y by 2.  Thus  Y  is eliminated.  The problem becomes easy from this point onwards.
     For another example of simultaneous equations, please refer to thisarticle.

HeuristicsUsed

H04. Look for pattern(s)

H09. Restate the problem in another way

H10. Simplify the problem

H11. Solve part of the problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* Lower Secondary Mathematics (Secondary 1)

* GCE ‘O’ Level “Elementary” Mathematics (revision)

* other syllabuses that involve algebra and simultaneous linear equations