Solving Two Variable System of Equations

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Solving Two Variable System of Equations
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Purpose of two system of equations

• When there are two sets of equations with the
  same set of unknown values, you can use system of
  equations to find the unknown value to solve both
  equations.

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You can solve these equations by...

• Substitution
• Elimination
• Matrices
• Graphing

4
When to use substitution

• The most efficient way to use substitution is
  when there is a variable that can easily be
  isolated. Pick the equation where a single
  variable can be isolated and plug it into the
  other equation.
• Ex
• 2x-3y-2
• 4xy24
• y24-4x
• Plug in y
• 2x-3(24-4x)-2
• x5
• Plug in x and solve for y

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When to use elimination

• By cancelling out one variable, it is possible to
  solve for the remaining unknown value. The
  coefficients must cancel out by multiplying one
  or both equations to achieve that. After, add the
  equations.
• Ex
• 2xy9
• 3x-y16
• 5x25
• x5 Solve for y. y-1

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Matrices

• To be more specific, cramer's rule is used when
  solving systems using matrices.
• This works for both 2 and 3 Variable systems of
  equations.
• For example
• 2x 3y z 10
• x - y z 4
• 4x - y - 5z -8

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Continuation

• Each unknown will be the quotient of the
  determinant obtained by substituting the answers
  in the right sides of the equations for the
  coefficients of the unknown divided by the
  determinant formed by taking the coefficients on
  the left sides of the equations.
• Lets try to solve for x only
• x 10 3 1 2 3 1
• 4 -1 1 divided by 1 -1 1
• -8 -1 -5 4 -1 -5

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Continuation

• We then find the determinant of the matrix that
  is the divisor.
• Det 2-1 1 - 31 1 11 -1
• -1 -5 4 -5 4 -1
• 2(51) - 3(-5-4) 1(-14)
• 12 27 3
• 42.
• Then we find the determinant of the matrix that
  is the dividend.

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Continuation

• Det 10-1 1 - 34 1 14 -1
• -1 -5 -8 -5 -8 -1
• 10(51) - 3(-208) 1(-4-8)
• 60 36 - 12
• 84.
• Since we have gotten both determinants
• x 84/42
• x 2.
• The same procedure is followed to find the
  remaining variables.

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Graphing

• This is by far the easiest way to solve a system
  of equations with two variables.
• The most common way of solving a system using
  graphing is by using a table of values,
• For example
• y-3x -2
• xy -6
• The solution to the systems is the point of
  intersection of the two graphs- (-1, -5).

x y
-1 0 1 2 -5 -2 1 4
x y
-1 0 1 2 -5 -6 -7 -8