Graphing Systems of Linear Equations and Inequalities

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Graphing Systems of Linear
Equations and Inequalities
Created by Kenny Kong HKIS 200
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Lesson Objectives

• Know that linear equations have graphs of
  straight lines.
• Know that the word line is part of the word
  linear.
• Know that an equation is linear without graphing
  the equation if

• The equation has no exponent greater than 1,

• The equation has no variable in the
  denominator, and

• The equation cant have the product of variables
  in the equation.

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Lesson Objectives Cont

• Know that a system is two or more equations with
  the same variables.
• Know that to solve for a system of linear
  equations is to determine the point(s) of
  intersection of the lines.
• Know that two lines can intersect in only one
  point, which means the system of linear equations
  has one solution.
• Know that when the lines do not intersect
  (parallel), there is no solution.

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Lesson Objectives Cont

• Know that if the lines coincide, (which means
  they are the same line), then there are an
  infinite number of solutions since every point on
  the line is a point of intersection.
• Know that a system of inequalities is two or more
  inequalities with the same variables.

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Lesson Objectives Cont

• Know that to graph systems of linear inequalities
  is like graphing systems of linear equations
  except that graph of an inequality consists of a
  boundary line and a shaded area.
• Know that the solution of the system of
  inequalities will be the intersection of the
  shaded area.

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What is a Linear Equation?

• A linear equation will always graph into a
  straight line.
• The word line is part of the word linear.
• Without graphing, an equation is a linear the
  equation if

• The equation has no exponent greater than 1,

• The equation has no variable in the
  denominator, and

• The equation cant have the product of variables
  in the equation.

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Examples of Linear Equations

• 3x y 17
• 4/9 x 2y 11
• 5x 6y 28
• x 8

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Examples of Equations that are Not Linear

• X2 y 9

• Equation contains a variable greater 1.

• Equation contains a variable in the denominator.

5/x 7

• Equation contains the product of two variables.

xy 24
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What is a System of Linear Equations?

• A system is two or more equations with the same
  variables.
• If you have two different variables, you need two
  equations.
• A system of linear equations has more than one
  equation, so its graph would be more than one
  line.
• To solve for a system of linear equations is to
  determine the point(s) of intersection of the
  lines.

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What is a System of Linear
Equations? Cont

• Since two lines can intersect in only one point,
  that means the system of linear equations has one
  solution.
• When the lines do not intersect (parallel), you
  have no solution.
• If the lines coincide, which means they are the
  same line, then there are an infinite number of
  solutions since every point on the line is a
  point of intersection.

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Case 1

• The two lines can intersect in only one point,
  that means the system of linear equations has one
  solution.

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Case 2

• The lines do not intersect (parallel), you have
  no solution (?).

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Case 3

• The lines coincide. When the lines coincide, you
  have an infinite number of solutions.

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Graphing a System of Linear Equations

• To graph a system of linear equations, you will
  use the slope-intercept form of graphing.
• The first step is to transform the equations into
  slope-intercept form or y mx b.
• Then use the slope and y-intercept to graph the
  line.
• Once you have both lines graphed, determine your
  solutions.

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Example x y 6 2x y 3

• Transform the first equation into y mx b

• x y 6

• Subtract x from both sides of the equation.

x y 6
x
x

• Simplify both sides of the equation.

- y 6 x

• Multiply by 1 to both sides of the equation.

- y 6 x
-1 ?
-1 ? ( )

• Simplify both sides of the equation.

y -6 x

• Use the commutative property.

y x -6
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Example x y 6 2x y 3
cont

• Transform the second equation into y mx b.

• 2x y 3

2x y 3
2x
2x

• Subtract 2x from both sides of the equation.

y 3 2x

• Simplify both sides of the equation.

• Use the commutative property.

y -2x 3
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Graphing x y 6 2x y 3

• In the first equation, y-intercept is -6.

• From that point, go up 1 and right 1 because
  slope is 1.

Point of intersection is the solution (3,-3)

• Draw a line through the y-intercept and the
  endpoint.

• the second equations, y-intercept is 3.

• From that point, go down 2 and to the right 1.

• Draw a line through the y-intercept and the
  endpoint.

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Heres a time-saver

• The nature of the solutions of a system can be
  determined without actually graphing them.
• Once you have the equations transformed into
  y mx b, compare the slopes and y-intercepts
  of each equation.
• If the slopes and y-intercepts are the same, you
  will have the same line.
• If the slopes are the same, but the y-intercepts
  are different, you will have parallel lines.
• If the slopes are different, then the lines will
  intersect.

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Case 1

• The slope and y-intercepts are the same, ? the
  lines coincide. There will be an infinite number
  of solutions.
• y 3x 4
• y 3x 4

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Case 2

• The slopes are the same, but the y-intercepts are
  different, so the lines are parallel. There is no
  solution.
• y 3x 6
• y 3x 6

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Case 3

• The slopes are different, so the lines will
  intersect, so there will be one solution.
• y 3x 5
• y 4x 6

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Solving Systems of Inequalities Graphically

• A system of inequalities is two or more
  inequalities with the same variables.
• You graph systems of linear inequalities like you
  graph systems of linear equations except that
  graphing of an inequality consists of a boundary
  line and a shaded area.

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To graph a system of inequalities

• Transform the inequality into y mx b.
• Graph the boundary line.
• Then determine if you will shade above or below
  the boundary line.
• The solution of the system of inequalities will
  be the intersection of the shaded areas.

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Example y gt x y lt 3

• The inequalities are already in y mx b form.

• Start with the y-intercept of 0.

• Then go up 1 and to the right 1 because the slope
  is 1.

• The boundary line will be dotted because the
  inequality symbol is gt.

• Because the inequality is gt, you will shade above
  the line.

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Example y gt x y lt 3

• The inequality y lt 3 will have a horizontal
  boundary line.

• The boundary line will be dotted.

• Shade below the line because the inequality
  symbol is lt.

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Example y gt x y lt 3

• The intersection of the two shaded areas is the
  solution of the system.

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Example y gt x y lt 3