This week’s blog post is a little different than the past few weeks. It is a little less math-intensive, but still very relevant. I will introduce a concept in mathematics that extends from Algebra and Geometry into higher math. It is a concept that eludes some of my best tutoring students. It is the concept of *The Substitution Principle*.

Basically, the substitution principal states in plain English that:

**If quantities have the same value then they are interchangeable.**

Sometimes the wording is different or the concept is clouded by the details of the problem, but it merely states that if two quantities are equal, in numeric or algebraic value, then one can take the place of the other in your calculation or mathematical proof.

Think about it in terms of an equation:

**Example 1:**

Given: x+ y = **z**

**z**=5

Conclusion: substitute 5 in for z into the top equation and get

x+y=5

**Example 2: **

a=b and a=f

We can conclude that **b=f**

**Example 3 (from Geometry):**

A = πr^{2}

A=18πy

We can conclude that πr^{2} =18πy and we can solve for r or y.

**Example 4 (from a geometric proof):**

Given: <1 complementary to <3 (i.e. m<1 + m<3 = 90)

<2 complementary to <3 (i.e. m<2 + m<3 = 90)

Conclusion: m<1 + m<3 = m<2 + m<3

m<1 = m<2

As you can see, this principal has many useful applications.

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There is one huge caveat with the substitution principle — identifying “quantity” in the textual form of an expression. Basically, a substring represents a quantity if and only if you can surround it with parentheses without changing the meaning of the whole expression.

Good post Mike. This should be understood by most students that you tutor.

Dijkstra called this Leibniz’s rule in “Program Semantics and Predicate Calculus” (which I highly recommend, by the way). It’s never given enough explicit mention, since to an experienced practitioner it seems blindingly obvious. Good on you for bringing it up.

Thanks for the positive feedback, Bob. I will try some good and new topics in coming weeks.

Thank you . Sometimes the obvious can be overlooked at times. It is amazing the number of HS math students who find the principle foreign.

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I never thought I would agree with this opinion, but I’m starting to see things differently.

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Michelle

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