Linear Equations in Two Variables

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• Linear Equations in Two Variables

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Linear Equations in Two Variables

• may be put in the form
  
• Ax By C,
• Where A, B, and C are real numbers and A and
  B are not both zero.

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Solutions to Linear Equations in Two Variables

• Consider the equation
• The equations solution set is infinite because
  there are an infinite number of xs and ys that
  make it TRUE.
• For example, the ordered pair (0, 10) is a
  solution because
• Can you list other ordered pairs that satisfy
  this equation?

Ordered Pairs are listed with the x-value first
and the y-value second.
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Input-Output Machines

• We can think of equations as input-output
  machines. The x-values being the inputs and the
  y-values being the outputs.
• Choosing any value for input and plugging it
  into the equation, we solve for the output.

y -2x 5 y -2(4) 5 y -8 5 y -3
x 4
y -3
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Functions

• Function- a relationship between two variables
  (equation) so that for every INPUT there is
  EXACTLY one OUTPUT.
• To determine (algebraically) if an equation is a
  function we can examine its x/y table. If it is
  possible to get two different outputs for a
  certain input- it is NOT a function. In this case
  an x-value in the table or ordered pairs would
  repeat.
• This may be determined (graphically) by using the
  Vertical Line Test. If any vertical line would
  touch the graph at more than one point- it is NOT
  a function.

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Using Tables to List Solutions

• For an equation
• we can list some solutions in a table.
• Or, we may list the
• solutions in ordered pairs .
• (0,-4), (6,0), (3,-2),
• ( 3/2, -3), (-3,-6),
• (-6,-8),

x y
0 -4
6 0
3 -2
3/2 -3
-3 -6
-6 -8

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Graphing a Solution Set

• To obtain a more complete picture of a solution
  set we can graph the ordered pairs from our table
  onto a rectangular coordinate system.
• Lets familiarize ourselves with the Cartesian
  coordinate system.

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Cartesian Plane




















y-axis
Quadrant II ( - ,)
Quadrant I (,)
x- axis
origin
Quadrant IV (, - )
Quadrant III ( - , - )
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Graphing Ordered Pairs on a Cartesian Plane




















y-axis

• Begin at the origin
• Use the x-coordinate to move right () or left
  (-) on the x-axis
• From that position move either up() or down(-)
  according to the y-coordinate
• Place a dot to indicate a point on the plane
• Examples (0,-4)
• (6, 0)
• (-3,-6)

(6,0)
x- axis
(0,-4)
(-3, -6)
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Graphing More Ordered Pairs from our Table for
the equation
y

• Plotting more points
• we see a pattern.
• Connecting the points
• a line is formed.
• We indicate that the
• pattern continues by placing
• arrows on the line.
• Every point on this line is a
• solution of its equation.

x
(3,-2)
(3/2,-3)
(-6, -8)
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Graphing Linear Equationsin Two Variables
y

• The graph of any linear equation in two variables
  is a straight line.
• Finding intercepts can be helpful when graphing.
• The x-intercept is the point where the line
  crosses the x-axis.
• The y-intercept is the point where the line
  crosses the y-axis.
• On our previous graph, y 2x 3y 12,
  find the intercepts.

x
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Graphing Linear Equationsin Two Variables
y

• On our previous graph, y 2x 3y 12,
  find the intercepts.
• The x-intercept is (6,0).
• The y-intercept is (0,-4).

x
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Finding INTERCEPTS

• To find the x-intercept Plug in ZERO for y
  and solve for x.
• 2x 3y 12
• 2x 3(0) 12
• 2x 12
• x 6
• Thus, the x-intercept is (6,0).

To find the y-intercept Plug in
ZERO for x and solve for y. 2(0) 3y 12
2(0) 3y 12 -3y 12 y -4 Thus,
the y-intercept is (0,-4).
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Special Lines

• y 5 0 x 3y -5

y




x





y





x



y is a horizontal line x is a
vertical line
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SLOPE

• SLOPE- is the rate of change
• We sometimes think of it as the steepness, slant,
  or grade.

Slope formula
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SlopeGiven 2 colinear points, find the slope.

• Find the slope of the line containing (3,2) and
  (-1,5).

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Slopes

• Positive slopes rise from left to right
• Negative slopes fall from left to right

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Special Slopes

• Vertical lines have UNDEFINED slope (run0 ---
  undefined)
• Horizontal lines have zero slope (rise 0)
• Parallel lines have the same slope (same slant)
• Perpendicular lines have opposite reciprocal
  slopes

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Slope-Intercept Form

• y mx b
• where m is the slope and b is the y-intercept

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Graph using Slope-Intercept form
Given 2y 6x 4 y 3x 2
Plot (0, -2) then use 3/1 as rise/run to
get 2nd point
y


1
x
3

b

• Solve for y.
• Plot b on the y-axis.
• Use
• to plot a second
  point.
• 4. Connect the points to make a line.

Rise positive means UP/ negative means DOWN Run
positive means RIGHT/ negative means LEFT
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Determine the relationship between lines using
their slopes
Same Slope Parallel Lines

• Are the lines parallel, perpendicular or neither?
  
• Solve for y to get in Slope-Intercept form.
• Then compare slopes.

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Determine the relationship between lines using
their slopes
Perpendicular Lines

• Are the lines parallel, perpendicular or neither?
  
• Solve for y to get in Slope-Intercept form.
• Then compare slopes.

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Write an Equationgiven the slope and y-intercept

• Given That a line passes through (0,-9) and has
  a slope of ½ , write its equation.
• (0,-9) is the y-intercept (because x0)
• ½ is the slope or m
• Plug into the Slope Intercept Formula to get
  y ½ x - 9

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Point-Slope Form

• At times we may not know the y-intercept. Thus,
  we need a new formula. The point-slope form of a
  line going through
• with a slope of m is given by

Use the Parentheses!
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Use Point-Slope when you dont have a y-intercept

• Given two points (1,5) and (-4,-2), write the
  equation for their line.
• Choose one point to plug in for (x1,y1)
• Find the slope using both points and the slope
  formula.
• Solve the equation for y.

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Modeling Data with Linear Equations

• Data can sometimes be modeled by a linear
  function.
• Notice there is a basic trend. If we place a line
  over the tops of the bars it roughly fits. Each
  bar is close to the line. Thus points on the line
  should estimate our data.
• Given the equation to the line we can make
  predictions about this data.

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Modeling Data with Linear Equations

• The number of U.S. children (in thousands)
  educated at home for selected years is given in
  the table. Letting x3 represent the year 1993,
  use the first and last data points to write an
  equation in slope-intercept form to fit the data.

y128x 204