Systems of Linear Equations

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Systems of Linear Equations

• Math 0099
• Section 8.1-8.3
• Created and Presented by Laura Ralston

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What is a system of linear equations?

• It is two or more linear equations with the same
  variables considered at the same time.
• Number of variables equals number of linear
  equations in the system
• Examples
• x y 4 x y 2
• x y -10 3x 4y 7
  

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What is the solution set to a system?

• The solution set to the system of linear
  equations is ALL ordered pairs that are solutions
  to both equations, that is, makes both equations
  TRUE at the same time.
• To decide whether an ordered pair is a solution
  to a system, substitute the values for x and y in
  both equations. If the results for both
  equations are true then the ordered pair is a
  solution to the system.

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Example Determine if the given point is a
solution to the system.

• x y 2
• 3x 4y 7
• (1, 1)
• (4, -2)

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Questions to Answer

• How do we find the solution, if there is one?
• Will there always be a solution to a system of
  linear equations?
• Can there be more than one solution?

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Methods for Solving a System of Equations

• Graphing - Section 8.1
• Substitution - Section 8.2
• Addition (Elimination)
• Section 8.3

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GRAPHING Procedure

1. Graph the first equation in the coordinate plane
2. Graph the second equation on the same coordinate
   plane
3. Record the coordinates of the point of
   intersection of the two graphs. This ordered
   pair is the solution to the system
4. Check solution in both equations.

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Three possibilities for solutions for a system

• NO SOLUTION
• Graphically, the lines would be parallel.
• Solving for x will result in a false statement
  with no variable remaining
• INCONSISTENT

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• ONE SOLUTION
• Graphically, the lines will intersect ONCE.
  Solution will be an ordered pair
• Solving for x will result in a numerical value
• CONSISTENT

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• INFINITE SOLUTIONS
• Graphically, the lines coincide (same line)
• Solving for x results in a true statement with no
  variable remaining
• DEPENDENT

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Examples

• x 2y 8
• 2x y 1

• y 2x 5
• 4x 2y -10

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Assignment

• Page 595 1-7 odd, 13-39 odd

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SUBSTITUTION

• Objective is to eliminate one of the variables so
  that a new equation is formed with just one
  variable
• Most useful when one of the equation is solved
  for one variable already OR if one of the
  variables has a coefficient of 1 otherwise, you
  get Fractions !!! Fractions !!! Fractions !!!
• Provides exact answers rather than estimations

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Substitution Steps

• Solve one of the given equations for either x or
  y, whichever is easier.
• Substitute the result from step 1 into the other
  given equation
• Solve for the remaining variable
• Substitute (back substitute) this solution into
  one of the ORIGINAL given equations

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Substitution steps continued ...

• Solve for the variable. Write final solution as
  an ordered pair (x, y)
• Check answer in both given equations. True
  statements indicate correct answers.

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Examples

• x y 3 y -3 2x
• y 2x 4x 2y 6
• y 4 3x
• Y-3x 6

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Assignment

• Page 603 1-41 odd

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ADDITION (ELIMINATION)

• The idea is to eliminate one of the variables
  from the system of linear equations.
• To do this, one of the variables must have
  coefficients that are opposites.
• Provides exact answers rather than estimated ones

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Addition (Elimination) Steps

• Write each equation in standard form (align like
  terms)
• If needed, multiply one or both equations by
  appropriate number(s) so that the coefficients on
  either x or y are opposites.
• Add the equations from step 2 together by
  combining like terms. This should result in an
  equation with one variable.
• Solve the equation from step 3.

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Addition steps continued..

• Back Substitute the solution from step 4 into
  either of the ORIGINAL given equation
• Solve for the other variable. Write final answer
  in an ordered pair (x, y)
• Check your answer in each original given
  equation. True statements result in correct
  answers.

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Examples

• 2x 2y 4
• x y -3
• y 3x 15
• 6x 2y -30

• 2x 5y 6
• 4x 10y -2

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Assignment

• Page 611 1-41 odd

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COMPASS Practice Questions

• What is the solution of the system of equations
  below?
• A. (3a, 2a)
• B. (-3a, 2a) 3x 4y a
• C. (15a, 11a) 2x 4y 14a
• D. (15a, -11a)
• E. (3a, -2a)

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• What are the (x, y) coordinates of the point of
  intersection of the lines determined by the
  equations 2x 3y 4 and y x?
• A. (4, 4) B. (4, 4) C. (4, 4)
  D. (4, 4) E. (2, 0)