Systems of Linear Equations in Two Variables

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Section 4.1

• Systems of Linear Equations in Two Variables

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What is a system of equations?

• A system of equations is a group of two or more
  equations, each of which contains one or more
  variables.
• A solution to a linear system is the value of
  the variable(s) that make ALL equations a true
  statement.

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Solutions to systems of 2 equations

• Systems of 2 linear equations represent 2 lines.
• What are our options when we have 2 lines?
• Option 1 Intersect
• How many solution points?
• (how many combinations of
• x and y are on both lines?)
• -This case- the system is called
• CONSISTANT and INDEPENDENT
• There is one unique solution.

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Systems of 2 linear equations

• Option 2 Parallel
• How many points are there that are on both lines?
  (that make both eqns true?)
• This system is called INCONSISTENT
• There is NO solution

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More systems options

• Option 3 They are the same line. (they coincide)
• How many points are there that are on both lines?
• This system is called CONSISTENT and DEPENDENT.
• There are infinitely many solutions (all points
  that lie on that line)

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Solve the system by graphing

• Remember to put into slope-intercept form to
  graph!

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Solving Systems by Substitution

• Graphing wont always work, especially if the 2
  lines dont intersect at integer values of x and
  y.
• Substitution is an easy method
• Solve one equation for one of the 2 variables.
• Substitute that expression into the other
  equation.

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Example- Substitution

• Solve 3x y 5
• -2x 5y 1

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Method of Elimination

• Basic idea get coefficients of one of the
  variables to be additive inverses (negatives of
  each other), then add equations.
• Multiply both sides of 1 or both equations by a
  NONZERO constant so that coefficients of one of
  variables will cancel.
• Add equations. One of the 2 variables drops out.
  Solve the resulting equation for the remaining
  variable.
• Substitute the value of the variable found in (2)
  into either original equation, solve for other
  variable.
• Check answer!

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Examples

• -x y 12
• 2x 3y -26
• 3x -5y -1
• 2x -4y -2

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Which method to use?

• Graphing If exact answers arent necessarily
  needed, just a visual.
• Substitution If one equation is already solved
  for one variable, or a coefficient of a variable
  is 1.
• Elimination If both equations are in Standard
  Form.

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More examples