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3'2 Solving Linear Systems Algebraically

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(used mostly when one of the equations has a variable with a coefficient of 1 or -1) ... Solve one of the given equations for one of the variables. ... – PowerPoint PPT presentation

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Title: 3'2 Solving Linear Systems Algebraically


1
3.2 Solving Linear Systems Algebraically
  • p. 148

2
2 Methods for Solving Algebraically
  • Substitution Method
  • (used mostly when one of the equations has a
    variable with a coefficient of 1 or -1)
  • Linear Combination Method

3
Substitution Method
  • Solve one of the given equations for one of the
    variables. (whichever is the easiest to solve
    for)
  • Substitute the expression from step 1 into the
    other equation and solve for the remaining
    variable.
  • Substitute the value from step 2 into the revised
    equation from step 1 and solve for the 2nd
    variable.
  • Write the solution as an ordered pair (x,y).

4
Ex Solve using substitution method
  • Now, solve for x.
  • 2x6x-26 -10
  • 8x16
  • x2
  • Now substitute the 2 in for x in for the equation
    from step 1.
  • y3(2)-13
  • y6-13
  • y-7
  • Solution (2,-7)
  • Plug in to check soln.
  • 3x-y13
  • 2x2y -10
  • Solve the 1st eqn for y.
  • 3x-y13
  • -y -3x13
  • y3x-13
  • Now substitute 3x-13 in for the y in the 2nd
    equation.
  • 2x2(3x-13) -10

5
Linear Combination Method
  • Multiply one or both equations by a real number
    so that when the equations are added together one
    variable will cancel out.
  • Add the 2 equations together. Solve for the
    remaining variable.
  • Substitute the value form step 2 into one of the
    original equations and solve for the other
    variable.
  • Write the solution as an ordered pair (x,y).

6
Ex Solve using lin. combo. method.
  • 2x-6y19
  • -3x2y10
  • Multiply the entire 2nd eqn. by 3 so that the ys
    will cancel.
  • 2x-6y19
  • -9x6y30
  • Now add the 2 equations.
  • -7x49
  • and solve for the variable.
  • x-7
  • Substitute the -7 in for x in one of the original
    equations.
  • 2(-7)-6y19
  • -14-6y19
  • -6y33
  • y -11/2
  • Now write as an ordered pair.
  • (-7, -11/2)
  • Plug into both equations to check.

7
Ex Solve using either method.
  • 9x-3y15
  • -3xy -5
  • Which method?
  • Substitution!
  • Solve 2nd eqn for y.
  • y3x-5
  • 9x-3(3x-5)15
  • 9x-9x1515
  • 1515
  • OK, so?
  • What does this mean?
  • Both equations are for the same line!
  • many solutions
  • Means any point on the line is a solution.

8
Ex Solve using either method.
  • 6x-4y14
  • -3x2y7
  • Which method?
  • Linear combo!
  • Multiply 2nd eqn by 2.
  • 6x-4y14
  • -6x4y14
  • Add together.
  • 028
  • Huh?
  • What does this mean?
  • It means the 2 lines are parallel.
  • No solution
  • Since the lines do not intersect, they have no
    points in common.

9
Assignment
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