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:
Given: x+ y = z
Conclusion: substitute 5 in for z into the top equation and get
a=b and a=f
We can conclude that b=f
Example 3 (from Geometry):
A = πr2
We can conclude that πr2 =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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