$$\DeclareMathOperator{cosec}{cosec}$$

# Number and Algebra

Syllabus Content 1.6

Simple deductive proof, numerical and algebraic; how to lay out a left-hand side to right-hand side (LHS to RHS) proof. The symbols and notation for equality and identity

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## Furthermore

This video on Deduction is from Revision Village and is aimed at students taking the IB AA Maths Standard level course

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Example: Show that $$\frac{1}{4} + \frac{1}{12} = \frac{1}{3}$$. Show that the algebraic generalisation of this is $$\frac{1}{m+1} + \frac{1}{m^2+m} \equiv \frac{1}{m}$$

LHS to RHS proofs require students to begin with the left-hand side expression and transform this using known algebraic steps into the expression on the right-hand side (or vice versa).

Example: Show that $$(x-3)^2+5 \equiv x^2-6x+14$$.

Students will be expected to show how they can check a result including a check of their own results.

Example, Show that the algebraic generalisation of the following is true.

$$\frac{1}{m+1} + \frac{1}{m^2+m} \equiv \frac{1}{m}$$

Let's start by finding a common denominator for the left-hand side of the equation:

$$\frac{1}{m+1} + \frac{1}{m(m+1)}$$ $$\frac{m + 1}{m(m+1)}$$

Now, let's simplify the expression:

$$\frac{1}{m}$$

It is now evident that the original equation:

$$\frac{1}{m+1} + \frac{1}{m^2+m} \equiv \frac{1}{m}$$

Is generally true, as the left-hand side simplifies to $$\frac{1}{m}$$.

Thus, the algebraic generalisation provided does hold true for all values of $$m$$.

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