Solving Pairs of Linear EquationsLots of Ways

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Section 5.2

• Solving Pairs of Linear EquationsLots of Ways!

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SOVLING SIMULATNEOUS LINEAR EQUATIONS
Consider the pair (or system) of linear equations
of the form
where x and y are the unknowns and the
coefficients a, b, p in the first equation and c,
d, q in the second equation are given
constants. A solution of this system is simply a
pair (x, y) of numbers that make both equations
true at the same time (simultaneously).
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POSSIBILITIES FOR THE SOLUTION TO A LINEAR PAIR
OF EQUATIONS
There are just three possibilities for the
solution to a linear pair of equations

• Exactly one solution. Graphically, this is two
  intersecting lines.
• No solution. Graphically, this is two parallel
  lines.
• Infinitely many solutions. Graphically, this is
  two coincident lines (two linesone on top of the
  other).

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GRAPHICAL METHOD FOR SOLVING

• Solve both equations for y that is, make each
  equation look like y something.
• Graph each equation (on your calculator).
• Use the intersection feature of your calculator
  to find the x and y that solve the system.

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ALGEBRAIC SOLUTION METHOD OF ELIMINATION

• Add an appropriate constant multiple of the first
  equation to the second equation. (The idea is to
  choose the constant multiple that serves to
  eliminate the variable x from the second
  equation.)
• Solve the resulting equation for y.
• Substitute the value for y back into one of the
  original equations to find x.

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SYMBOLIC SOLUTIONS
Given the system of equations
The solutions are
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DETERMINANTS
The fractions on the previous slide both have
denominators of ad - bc. This is the value of
the 2-by-2 (or 2 2) determinant defined by
NOTE ? is upper-case Greek letter delta.
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DETERMINANT SOLUTION TO A PAIR OF LINEAR EQUATIONS
Given the system of equations
The solutions are given by
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REMARKS ON DETERMINANT SOLUTIONS
Consider
The value for ? gives information about the
solutions to the system

• If ? ? 0, the system has exactly one solution.
• If ? 0, the system either
• has no solutions or
• has infinitely many solutions.

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MATRIX
Definition A matrix is an array (table) of
numbers. EXAMPLES
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MATRICES AND SYSTEMS OF EQUATIONS
The solution to the system of equations
is determined by its coefficient matrix A
and its constant matrix B.
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MATRICES AND SYSTMES(CONTINUED)
We can abbreviate the system
by writing
or simply
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SOLVING A SYSTEM OF LINEAR EQUATIONS USING
MATRICES
The solution to the system of equations
is given by the matrix equation