A & S 101-003
February 5 Class Notes
Equivalence Relations
Comment on Equivalence Equations from Dr. Jones
π During last class, the discussion was made that algebraic equations are equivalent
if their solution sets are identical. Equations can also have equivalence by having
the same domain. Equations with the same domain still satisfy the three
properties that equations with the same solution set do.
1) Reflexive property
2) Symmetric property
3) Transitive property
Dr. Seif said the he would make up a sheet with examples of when equivalence relations
came up naturally.
Unfinished business on equivalence relations: If you have an equation and multiply it by
a variable, you usually get two different solution sets. What equivalence relations do
these two equations have? For example:
Equation 1: x(x + 1) = 5x. If you multiply by 1/x, then you get the second
equation.
Equation 2: x + 1 = 5 The solution to this equation is x = 4, but this
answer does not give the only solution to equation
1.
Comments:
π The thing about this situation is when we multiply equation 1 by 1/x, we are
assuming x to be a variable, but x in equation 1 is considered to be an unknown.
π In equation 1, 0 is a possible solution, but when we multiply equation 1 by 1/x,
we are multiplying the equation by 1/x, where x is zero. This cannot happen.
π When we multiply equation 1 by 1/x we are changing the limits of the equation as
well.
π In this case, we are making the solution set of equation 1 smaller by multiplying
by 1/x.
π Sometimes when you multiply by an algebraic expression with a variable the
solution set gets larger.
When teaching your class about solving equations, you can show them examples of
multiplying an equation by an algebraic expression with variables where the solution set
of an equation is larger and smaller to help them understand about this topic.
*** Be sure you try to account for what you do to an equation. This will help with
mistakes that can be made. ***
What happens when you square both sides of an equation?
What happens when you take the square root of both sides of an equation?
February 5 Class Notes
Equivalence Relations
Comment on Equivalence Equations from Dr. Jones
π During last class, the discussion was made that algebraic equations are equivalent
if their solution sets are identical. Equations can also have equivalence by having
the same domain. Equations with the same domain still satisfy the three
properties that equations with the same solution set do.
1) Reflexive property
2) Symmetric property
3) Transitive property
Dr. Seif said the he would make up a sheet with examples of when equivalence relations
came up naturally.
Unfinished business on equivalence relations: If you have an equation and multiply it by
a variable, you usually get two different solution sets. What equivalence relations do
these two equations have? For example:
Equation 1: x(x + 1) = 5x. If you multiply by 1/x, then you get the second
equation.
Equation 2: x + 1 = 5 The solution to this equation is x = 4, but this
answer does not give the only solution to equation
1.
Comments:
π The thing about this situation is when we multiply equation 1 by 1/x, we are
assuming x to be a variable, but x in equation 1 is considered to be an unknown.
π In equation 1, 0 is a possible solution, but when we multiply equation 1 by 1/x,
we are multiplying the equation by 1/x, where x is zero. This cannot happen.
π When we multiply equation 1 by 1/x we are changing the limits of the equation as
well.
π In this case, we are making the solution set of equation 1 smaller by multiplying
by 1/x.
π Sometimes when you multiply by an algebraic expression with a variable the
solution set gets larger.
When teaching your class about solving equations, you can show them examples of
multiplying an equation by an algebraic expression with variables where the solution set
of an equation is larger and smaller to help them understand about this topic.
*** Be sure you try to account for what you do to an equation. This will help with
mistakes that can be made. ***
What happens when you square both sides of an equation?
What happens when you take the square root of both sides of an equation?
