Systems of Linear Equations with two variables

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Systems of Linear Equations with two variables
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An interesting problem

• A 600-seat movie theater charges 5.50 admission
  for adults and 2.50 for children.
• If the theater is full, and 1911 is collected,
  how many adults and how many children are in the
  audience?

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To solve a real-life problem like this, we need
to

• formulate it as a mathematical problem,
• solve the math problem, and
• interpret the solution of the math problem

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Let
x number of children in attendance
y number of adults in attendance
xy 600 2.50x5.50y 1911
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Observe that both equations are linear.

• We call such a set of equations a

System of linear equations with two variables
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Solving a system of linear equations with two
variables

• by substitution (Read page 186)
• by elimination (Read page 187)
• by graphing (Read page 182)

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Example Solve the system obtained for the movie
theater problem by substitution.
xy 600 2.50x5.50y 1911
x 463 y 137
We conclude that there are 463 children and 137
adults in attendance.
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Example Solve the system obtained for the movie
theater problem by elimination.
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Example Solve the system obtained for the movie
theater problem by graphing.
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Three possible cases when we solve a system of
linear equations with 2 variables

• Exactly one solution.
• Graphically, there is one point of intersection
• No solution
• Graphically, the lines never intersect
• Infinitely many solutions
• Graphically, the lines are equivalent