[S2_20151107SLFC] Simultaneous Linear Equations with Fractional Coefficients

Question 

Introduction

     Somebody said, “Dear Algebra, please stop asking me about your  x.  She is not coming back.  Don’t ask me  y!”

     Here we have here linear equations that appear to be more tricky than the usual fanfare.  This is because the coefficients of the unknowns  x  and  y  are fractions.  In school, students learn to solve these using the method of substitution and the method of elimination.  They tend to prefer to do it by the former method, because psychologically it seems easier to accept.  However, the latter matter is generally more effective and yields a shorter solution.  This question is like a fly trap for those who prefer the method of substitution.  You make either  x  or  y  the subject from one of the equations, and then substitute that into the other equation.  This yields a complicated algebraic fractions within algebraic fractions.  The more complicated your solution is, the higher chance there is for making careless mistakes.  It is a good idea that students get out of their comfort zones and adopt a new skill.  So how do we do it?

The smart tactic

     First, we need to clear away the fractions by multiplying through with the LCM of the denominators appearing in each equation.  For the first equation, LCM(4, 8) = 8.  Since  4  is a factor of  8,  8  is like a giant that absorbs the number  4.  OK, so we multiply the first equation through by  8.  For the second equation, we multiply every term by LCM (3, 2) = 6

Solution

Remarks

     As you can see, equations [3] and [4] both contain – 3y  and this can be eliminated via subtraction.  So the value of  x  comes out easily.  Now once  x  is known, one can find the  y!

For another example of simultaneous equations, please refer to this article.

Heuristics Used

H04. Look for pattern(s)

H05. Work backwards

H10. Simplify the problem

H11. Solve part of the problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* Lower Secondary Mathematics (Secondary 2)

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

* other syllabuses that involve algebra, linear simultaneous equations

[S1_AFMLCM_20150412] Conquering Algebraic Fractions

Question

Introduction

     This is an equation involving algebraic fractions, usually for secondary 1 (approximately grade 7) pupils in Singapore.  Many students (and teachers?) like to use the “cross-multiplying” method, as shown in Solution 1.  A usually more efficient method is to multiply every term by the Lowest Common Multiple (LCM) of all the denominators appearing in the equation, as shown in Solution 2.

Discussion

     Note that division by zero is not allowed.  Furthermore, in algebra, it is dangerous to cancel or divide by an unknown quantity, because there is a possibility that you are dividing by zero.  So any division or cancellation by an unknown quantity must be justified beforehand.  Mathematics is not a game of blind senseless manipulations.  If you look at the second solution, which is short and sweet (only 4 steps), multiplying through by the LCM of denominators not only avoids this awkwardness, but it clears all the fractions in one fell swoop.  The solution takes only  4  steps, and it is in fact the recommended method.  All students, whether “good” or “poor” in maths, should use the second method.  Teachers who refuse to use/teach the LCM method (out of habit, or because their own teachers taught them otherwise, or because this makes them or their pupils “uncomfortable”) are really doing the weaker students a huge disservice.  You are widening the achievement gap.  The better students are better, precisely because they use better methods.  The longer one’s working is, the higher the chances of making mistakes and the more time is wasted.  If the “weaker” pupils have to jump through lots of hoops to achieve a certain standard before they are allowed to learn this “advanced” method (actually it’s just the normal method), they will have to unlearn their old method and may get confused as they learn this method.  A triple whammy!  All learners need to practice anyway, so one might as well practice the correct thing right from the beginning and learn good habits (striving for efficient, effective, elegant solutions).  So please, please, please everyone: use the LCM method!

Suitable Levels

* Secondary 1 Mathematics

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

* other syllabuses that involve algebraic Fractions

* precocious children who want to learn algebra