Simultaneous Equations

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Simultaneous Equations

• Graphical Solutions
• Solution by Addition or Subtraction
• Solution by Substitution
• Determinants

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Graphical Solutions
Two linear equations plotted on the same coordinate axes will intersect at a single point Unless The graphs are parallel The graphs are co-linear

2x - 3y 2 4x 6y 12 2x 3y -6 x 2y 8 2x 3y -6 -x 3/2 y 3 2x - 3y 2 4x 6y 12 2x 3y -6 x 2y 8 2x 3y -6 -x 3/2 y 3
Lines intersect at one point only Lines are parallel Lines coincide Exactly one solution x4, y2 No solution Infinite solutions Lines intersect at one point only Lines are parallel Lines coincide Exactly one solution x4, y2 No solution Infinite solutions

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Graphical Solutions (2)
Solve
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Solution by Addition or Subtraction

• Put equations in the form
• (1) a1x b1y c1
• (2) a2x b2y c2
• Multiply the equations values that will produce
  the same coefficient for x or y.
• Add or subtract the equations to remove one of
  the variables
• Solve
• 7 y 3x 3
• 5 x 2 y

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Solution by Substitution

• Put equations in the form
• y a1x b1 or x a1y b1
• Replace one variable with its equivalent
  expression
• Solve for the remaining variable
• Solve
• y 0.4x
• y 50 .3x

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Determinants

• A determinant is the value computed from a square
  matrix of numbers by a rule of combining products
  of the matrix entries
• The number of rows (or columns) is called the
  order of the determinant.
• The diagonal, from the upper left corner to the
  lower right corner, is called the principal
  diagonal.
• The diagonal from the lower left corner to the
  upper right corner is called the secondary
  diagonal.

ad -bc
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Determinants (2)

• Using Determinants to solve linear equations
• Put equations in the form
• (1) a1x b1y k1
• (2) a2x b2y k2

The denominator for both unknowns is a1b2 a2b1 The numerator of x is k1b2 k2b1 The numerator of y is a1k2 a2k1