Chapter%208:%20Linear%20Equations%20as%20Models

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Chapter 8 Linear Equations as Models

• Advanced Math

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Section 8.1 Linear Growth and Decay

• Slope Intercept Form of an Equation
• y mx b m slope b y intercept
• Slope
• rise over run
• change in y over the change in x

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• Without graphing, find the slope and the vertical
  intercept of the line modeled by each equation.
• Y 7x 4
• Slope 7
• Y intercept 4
• Y ½x 17
• Slope ½
• y intercept 17
• Try these on your own
• Y 9 4x
• Slope 4
• Y-intercept 9
• Y 7/6x 2
• Slope 7/6
• Y-intercept 2

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Sample 1

• When you travel up from Earths surface, the air
  temperature decreases by about 11 degrees for
  each mile you rise about the ground. Suppose the
  temperature of the air at ground level is 68
  degrees.
• Model the situation with a table of values.
• Use your table to make a graph. Find the slope
  and the vertical intercept of the line that
  contains the points. Explain what they mean in
  terms of the situation.
• Write an equation for the temperature of the air
  (t) as a function of the height in miles above
  the ground (h).

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Try this one on your own

• Frankie has 500 in her savings account at the
  beginning of the year. Each month she saves 150.
• Model the situation with a table of values and a
  graph.
• Find the slope and the vertical intercept of the
  line. Explain what they mean in terms of the
  situation.
• Write an equation for the savings (s) as a
  function of the time in months (t).

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Linear Growth and Decay

• Linear Growth
• Same Value Repeatedly Added
• Positive Slope

• Linear Decay
• Same Value Repeatedly Subtracted
• Negative Slope

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Sample 2

• Classify each situation as a linear growth,
  linear decay, or neither. Explain your choice.
• Eleanor Basave put 5 in a piggy bank for her
  granddaughter when she was born. Every day after
  that Eleanor put a quarter in the bank.
• Linear Growth
• During each year, the shortest distance between
  Earth and the sun is 94,000,000 miles and the
  greatest distance is 94,500,000.
• Neither
• A teacher on a remote Arctic island buys 144 cans
  of fruit for the year. Each week the teacher eats
  4 cans of fruit.
• Linear Decay

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Section 8.2 Linear Combinations

• A solution of an equation with two variables is
  an ordered pair of numbers that makes the
  equation true.
• All the points whose coordinates are solutions of
  an equation form the graph of the equation.

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Standard Form of an Equation for a Line

• The equation 8t 12d 48 is called a linear
  equation because it is an equation of a line.
• Here is the standard form of a linear equation
• ax by c

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Sample 1

• Several times a week, Dynah Colwin runs part of
  the way and walks part of the way on a trail that
  is 4 miles long. Her running speed is 6 miles per
  hour and her walking speed is 4 miles per hour.
• Write an equation for this situation.
• Let r the time in hours Dynah Colwin spends
  running
• Let w the time in hours she spends walking
• Rewrite the equation in slope-intercept form.

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Try this one on your own

• Each week, Will swims the backstroke part of the
  way and the crawl part of the way for 30 laps of
  the swimming pool. His backstroke speed is 3 laps
  per minute and his crawl speed is 2 laps per
  minute.
• Write an equation for this situation.
• Let b time it takes to swim the backstroke
• Let c time it takes to swim the crawl
• Rewrite the equation in slope-intercept form.

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Intercepts of a Line

• Vertical Intercept (y intercept)
• x 0
• Horizontal Intercept ( x intercept)
• y 0

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Sample 2

• Find the intercepts of the graph 2x 5y 15.
  Use them to graph the equation.

• Try this one on your own
• Find the intercepts of the graph of
  y -2x 2. Use them to graph the equation.

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Section 8.3 Horizontal and Vertical Lines

• When you have a horizontal line, the slope is
  zero.
• The coefficient of x will be zero.
• When you have a vertical line, the slope is
  undefined.
• The coefficient of y will be zero.

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Sample 1

• Find the slope of each line and write an equation
  for each line.
• A
• The slope is 0.
• The y-coordinate is always 2.5.
• Y 2.5
• B
• The slope is undefined.
• The x-coordinate is always -1.
• X -1

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Try these on your own

• Find the slope of each line and write an equation
  for each line.
• A
• Slope 0
• y -7.6
• B
• Slope undefined
• x 1.2

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Sample 2

• Write an equation for each line.
• The point (-2,4) and (-2,0) are on the line.
• X-coordinates are both -2.
• Every point on the line has the same x
  coordinate, -2.
• X -2
• The slope is zero, and the point (2,3) is on the
  line.
• Slope is zero.
• Every point on the line has the same y
  coordinate, 3
• Y 3

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Try these on your own

• Write an equation for each line.
• The point (9,3) and (-1,3) are on the line.
• y 3
• The slope is undefined, and the point (-1,4) is
  on the line.
• x -1

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Section 8.4 Writing Equations for Lines

• Estimate an equation for the fitted line for the
  Olympic data.
• Step One Find the slope.
• Step Two Find the y-intercept.
• Step Three Re-write equation (y mx b)
  filling in the slope and y-intercept.

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Try this one on your own

• Estimate an equation for the fitted line for the
  data. Interpret Week 0 to mean the beginning of
  the first week.
• Slope 4
• y intercept 15
• y 4x 15

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Sample 2 Exact Equation from Two Points

• Two points on a line are (4,3) and (-6,-2). Write
  an equation for the line.
• Step One Find the slope.
• Step Two Find the y-intercept.
• Step Three Re-write equation (y mx b)
  inputting the slope and y-intercept.

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Try this one on your own

• Two points on a line are (1, -2) and (5,4). Write
  an equation for the line.
• Step One Find the slope.
• slope 3/2
• Step Two Find the y-intercept.
• y intercept -7/2
• Step Three Re-write equation.
• y 3/2x 7/2

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Sample 3 Equations from Other Facts

• The slope of a line is 3/2. One point on the line
  is (-4,1). Write an equation for the line.
• Two Methods to Find the Y-Intercept
• Substitution
• Use the Slope-Intercept Formula
• Graph

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Try this one on your own

• The slope of a line is -1/2. One point on the
  line is (3,4). Write an equation for the line.

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Section 8.5 Graphing Systems of Linear Equations

• MoviesPlus rents videos for 2.50 each and has no
  membership fee. Videobusters rents videos for 2
  each but has a 10 membership fee.
• Write and solve a system of equation to model
  this situation. (Two Options Graphing and
  Setting Them Equal)
• Let c total cost of renting videos
• Let n number of videos rented
• What advice would you give to someone trying to
  decide which video store to use?

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Try this one on your own

• The service charge on a checking account at
  Hometown Bank is 5 per month plus 0.15 for each
  check written. The service charge at Twentieth
  Century Bank is 0.25 per check.
• Write and solve system of equations to model this
  situation.
• Let c cost of the service charge
• Let n number of checks written
• What advice would you give to someone trying to
  decide which bank to use?

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Sample 2 Systems without Solutions

• Sandy and Rita are practicing for 100 m dash
  competition. Sandy gives Rita a 10 m head start.
• Write and graph a system of equations to model
  this situation.
• Let x time since the runner started
• Let y runners distance from starting line

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Try this one on your own

• James left the campground riding his bicycle at
  15 miles per hour. When he was 5 miles away from
  camp, Lucie left the camp riding at the same
  speed.
• Write and graph a system of equations to model
  this situation.
• Let x time Lucie started
• Let y their distance from the camp

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Section 8.6 Graphing Linear Inequalities

• Linear Inequalities on a Coordinate Plane
• The graph of a linear inequality is a region on a
  coordinate plane whose edge is a line.
• The line is called a boundary line.

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Sample 1 Graph each inequality.
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Try these on your own
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Talk it Over

• Tell whether the boundary line for each
  inequality is solid or dashed. Then tell whether
  you would shade the region above or below the
  line.

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Sample 2

• Derek has saved 48 by baby-sitting. He plans to
  use the money to buy tapes and compact discs at a
  music store. Tapes cost 8 and compact discs cost
  12. The inequality represents the amount
  Derek can spend at the store.
• Graph the inequality

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Try this one on your own

• Graph the inequality 5y 3x gt 15.
• Dashed or Solid Line?
• Dashed
• Shade Above or Below?
• Below

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Sample 3 Graphing Inequalities in Standard Form

• Graph the inequality 8x 12y lt 48.
• Use the intercepts to graph the boundary line.
• Dashed or Solid?
• Check test point.

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Try this one on your own

• Graph the inequality