8.1 Solving Systems of Linear Equations by Graphing

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8.1 Solving Systems of Linear Equations by
Graphing

• To solve by graphing, graph both linear
  equations. This gives an approximate solution.
  Algebraic methods are more exact (next 2
  sections).
• If the graphs intersect at one point the system
  is consistent and the equations are independent.

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8.1 Solving Systems of Linear Equations by
Graphing

• If the graphs are parallel lines, there is no
  solution and the solution set is ?. The system is
  inconsistent.
• If the graphs represent the same line, there are
  an infinite number of solutions. The equations
  are dependent.

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8.2 Solving Systems of Linear Equations by
Substitution

• Solving by substitution
• Solve for a variable
• Substitute for that variable in the other
  equation
• Solve this equation for the remaining variable
• Put your solution back into either of the
  original equations to solve for the other
  variable
• Check your solution with the other equation

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8.2 Solving Systems of Linear Equations by
Substitution

• ExampleFrom the first equation we get y2x-7,
  so substituting into the second equation

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8.2 Solving Systems of Linear Equations by
Substitution

• If when using substitution both variables drop
  out and you get something like 106The system
  inconsistent and there is no solution (parallel
  lines)
• If when using substitution both variables drop
  out and you get something like 1010The system
  dependent and every solution of one line is also
  on the other (same lines)

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8.3 Solving Systems of Linear Equations by
Elimination

• Solving systems of equations by elimination
• Write equations in standard form (variables line
  up)
• Multiply one of the equations to get coefficients
  of one of the variables to be opposites
• Add (or subtract) equations so that one
  variable drops out
• Solve for the remaining variable.
• Plug you solution back into one of the original
  equations and solve for the other variable.

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8.3 Solving Systems of Linear Equations by
Elimination

• Example
• Multiply the second equation by 3 to get
• Adding equations you get

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8.3 Solving Systems of Linear Equations by
Elimination

• If when using elimination both variables drop out
  and you get something like 106The system
  inconsistent and there is no solution (parallel
  lines)
• If when using elimination both variables drop out
  and you get something like 1010The system
  dependent and every solution of one line is also
  on the other (same lines)

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8.4 Linear Systems of Equations in Three Variables

• Linear system of equation in 3 variables
• Example

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8.4 Linear Systems of Equations in Three Variables

• Graphs of linear systems in 3 variables
• Single point (3 planes intersect at a point)
• Line (3 planes intersect at a line)
• No solution (all 3 equations are parallel planes)
• Plane (all 3 equations are the same plane)

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8.4 Linear Systems of Equations in Three Variables

• Solving linear systems in 3 variables
• Eliminate a variable using any 2 equations
• Eliminate the same variable using 2 other
  equations
• Eliminate a different variable from the equations
  obtained from (1) and (2)

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8.4 Linear Systems of Equations in Three Variables

• Solving linear systems in 3 variables
• Use the solution from (3) to substitute into 2 of
  the equations. Eliminate one variable to find a
  second value.
• Use the values of the 2 variables to find the
  value of the third variable.
• Check the solution in all original equations.

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8.5 Applications of Linear Systems of Equations

• Solving an applied problem by writing a system of
  equations
• Determine what you are to find assign variables
• Draw a diagram, figure or make a chart of
  information.
• Write the system of equations
• Solve the system using substitution or
  elimination
• Answer the question from the problem.

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8.5 Applications of Linear Systems of Equations

• Mixture problem How many ounces of a 5 solution
  must be added to a 20 solution to get 10 ounces
  of 12.5 solution.Let x ounces of 5
  solutionLet y ounces of 20 solution

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8.5 Applications of Linear Systems of Equations

• Solution to mixture problem in 2 variables