Chapter 4 Systems of Linear Equations

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Chapter 4Systems of Linear Equations
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What is a system of linear equations?

• Def A system of linear equations is two or more
  linear equations in two or more unknowns.
• For our purposes we will be only considering
  systems of linear equations that are two
  equations in two unknowns.
• Example
• 5x 9y 2
• 3x 7y 1

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What is a Solution of a System?

• Def A solution of a system of linear equations
  is an ordered pair that is a solution of both
  equations.
• Example
• The ordered pair ( -1, 3) is a solution of the
  system
• 3x y 0
• 2x 3y 7

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Example

• Determine if the ordered pair (-2, 5) is a
  solution of the system of linear equations.
• 5x - 2y 12
• 3x 4y 1

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Solving Systems of Linear Equations by Graphing

• Graph the first equation.
• Graph the second equation on the same axes.
• If the lines representing the two graphs
  intersect at a point, determine the coordinates
  of the intersection. The ordered pair is the
  solution to the system.
• Check the solution in both equations.

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Solve the system by graphing.

• 3x 2y 12
• 2x y 1

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Graph the first line.

• First, find the x and y intercepts of 3x 2y
  12.
• x-intercept y-intercept
• Let y 0 Let x 0
• 3x 2(0) 12 3(0) 2y 12
• 3x 12 2y 12
• x 4 y 6
• The x-intercept is 4 and the y-intercept is 6.

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Graph the first line.
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Graph the second line.

• First, find the x and y intercepts of 2x - y 1
• x-intercept y-intercept
• Let y 0 Let x 0
• 2x - (0) 1 2(0) - y 1
• 2x 1 -y 1
• x 0.5 y -1
• The x-intercept is 0.5 and the y-intercept is -1.

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Graph the second line on same axes.
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Find coordinates of intersection.
Point of intersection (2,3)
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Check the solution in both equations.

• Equation 1
• 3x 2y 12
• 3(2)2(3) ? 12
• 6 6 ? 12
• 12 12
• True

• Equation 2
• 2x - y 1
• 2(2) - (3) ? 1
• 4 - 3 ? 1
• 1 1
• True

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Number of Solutions to a Linear System of Two
Equations
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Solve the system by graphing.

• y 2x 3
• y 2x 7

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Graph the first line.

• Since the line, y 2x 3, is in
  slope-intercept form, we know the slope and
  y-intercept.
• The y-intercept is -3, so the line passes
  through (0, -3). The slope is 2/1. To graph, we
  start at the y-intercept and move 2 up (the rise)
  and 1 right (the run).

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Graph the first line.

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Graph the second line.

• Since the line, y 2x 7, is in
  slope-intercept form, we know the slope and
  y-intercept.
• The y-intercept is 7, so the line passes through
  (0, 7). The slope is 2/1. To graph, we start at
  the y-intercept and move 2 up (the rise) and 1
  right (the run).

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Graph the second line on same axes.
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Inconsistent System, No Solution.
Parallel Lines, Same Slope.
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Solve the system by graphing.

• y 2x 3
• 4x 2y 6

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Reference

• Beginning and Intermediate Algebra
• Blitzer
• 2nd Edition