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EquationSolving Continued
A+S 101003: High school mathematics from a more advanced
point of view
Goals of today’s discussion:
1. Briefly discuss the following. Yea or Nay?: Each
seminar two students (rotating
basis) take notes during the seminar and during the week summarize
the discussion.
2. Continue discussion concerning an algorithm for solving
linear equations in
one variable: why is that standard algorithm valid?
3. Which operations on an (arbitrary) equation lead to an
equivalent equation ?
Discussion.
4. Solving linear equations in one variable in other
algebraic systems : linear
congruences. Compare with linear equations in the real numbers .
5. Discuss quadratic equations in the real numbers:
develop an algorithm(s) to
solve quadratic equations.
6. What higher mathematics (studied in Abstract Algebra,
for example) is
relevant to these discussions of equationsolving?
7. Discuss systems of (linear) equations and algorithms to solve such systems.
8. Time permitting, connect linear systems and matrix equations.
At the end of the seminar last week, students were asked
to think about a
couple of questions. Summarized, these were:
Given an equation E, of the form A(t) = B(t), involving
one variable t , does
adding an algebraic expression involving the variable t to both sides result in
an
equivalent equation? Same question can be asked with ”multiplying” in place of
”adding”.
A challenge put forth by Dr. Jones: how can the problem
above be restated
so that it could be presented to a middle school or highschool student?
We’ll return to the questions after we go through a
standard algorithm for
solving a linear equation in one unknown (in the real numbers). Let’s do the
algorithm on this equation, pausing to reflect on our goals and justification
for the
steps in the procedure . Can we identify natural “subalgorithms” in the
procedure?
2x + 1 = 5x − 4
What about .2(3x − 4) = x/2 + 1? How would have a student begin?
We sadly (?) leave algebra in the real numbers and take up
the equation
2x+1 = 5x−4 in Z_{12}, the integers mod 12. Let’s recall some of
the basics there.
1. Z_{12} has twelve elements {0, 1, . . . , 10, 11}.
2. We add and multiply ”mod 12”. All questions below refer
to Z_{12}: What
is 3 × 5? What is 9 + 6? What is the additive identity? What is −5, the
additive inverse of 5? What is the multiplicative identity ?
3. Addition and multiplication Z_{12} are associative.
Let’s see what happens if we run on our algorithm on 2x +
1 = 5x − 4 but in
Z_{12}.
Let’s reflect on what happened.
Write a story problem whose solution is modelled by a linear equation in Z_{12}
Suggestion: something involving clocks and hours.
Equivalent systems of linear equations in the real numbers
Consider the following system of two equations in two
unknowns:
2x − 5y = 4
4x + y = 2
What do you we mean by a solution to the above system?
Assuming we are going to solve the system algebraically,
say with pencil and
paper, how do you proceed?
What does it mean it that two linear systems equivalent system of linear equations?
Quadratic equations in the real numbers
The algorithm for solving a linear equation with one
variable calls for isolating the
variable. Consider the equation
x^{2} = −2x + 5.
Is isolating the variable helpful here? Explain.
Consider the equation (x + 1)^{2} = 4. How might we solve it?
Now solve x^{2} = −2x + 5.
We continue the discussion (assuming time permits)– our
goal is an algorithm
for quadratic equations in the real numbers.
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