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Chapter 2 Section 3: Lines. In this section, we will… Find the slope of a line Identify the slope and the y -intercept of a line Graph lines given a point on the line and its slope Determine if given lines are parallel, perpendicular or neither

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slide1

Chapter 2 Section 3: Lines

  • In this section, we will…
  • Find the slope of a line
  • Identify the slope and the y-intercept of a line
  • Graph lines given a point on the line and its slope
  • Determine if given lines are parallel, perpendicular or neither
  • Find the equation of a line with given properties
  • Solve applications of linear equations
slide2

The General Form (or Standard Form) of a linear equation in two variables is any equation in the form: Ax + By = C where A, B, C integers.

Example:Circle each of the following equations that are linear equations.

Example:Circle each of the following graphs that are linear.

2.3 Linear Equations

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The slope(rate of change) of a line gives the line’s steepness.

Examples:Find the indicated slope.

We will leave the slope as an improper fraction

  • from A to B
  • from B to A
  • from A to C

Note:

2.3 Slope Definition and Examples

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Examples:Find the slope of the line connecting the points.

(3, 7) and (5, 4)

(-6, -9) and (-4, 1)

Examples:A line passes through the point (-2, -3) and has a slope of -2. Give three additional points on the line.

2.3 Calculating Slope

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Examples:Find the slope of the line connecting the points.

(5, -1) and (5, 7)

(6, -2) and (-3, -2)

Rule: Rule:

2.3 Calculating Slope

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Example:Find the rate of change of the line graphed.

Example:Next to each graph, write the letter of the description that best describes the slope.

  • Positive
  • Negative
  • Zero
  • Undefined

2.3 Slope Definition and Examples

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Example: Graph the line that

passes through the point (-4, 2)

with a slope of

Example: Graph the line that

passes through the point (5, -3)

with a slope of -4

2.3 Graphing Using the Slope

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Example: Graph the line that

passes through the point (-3, -2)

with a slope of 0

Example: Graph the line that

passes through the point (0, 3)

with a slope of undefined

2.3 Graphing Using the Slope

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The Slope-Intercept Form of a linear equation in two variables is any equation in the form: y = mx + b where m is the slope of the line and (0, b) is the coordinate of the y-intercept of the line.

  • What is the slope of the line?
  • What is the coordinate of the y-intercept of the line?
  • Write the equation of the line in slope-intercept form.

2.3 Identify the Slope and y-Intercept of a Line

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Example:Find the slope and the coordinate of the y-intercept of the given linear equation. Graph the equation using that information.

slope:

y-int:

2.3 Graph Linear Equations from Slope-Intercept Form

slide11

Example:Find the slope and the coordinate of the y-intercept of the given linear equation. Graph the equation using that information.

slope:

y-int:

2.3 Graph Linear Equations from Slope-Intercept Form

slide12

Example:Find the slope and the coordinate of the y-intercept of the given linear equation. Graph the equation using that information.

slope:

y-int:

2.3 Graph Linear Equations from Slope-Intercept Form

slide13

Example:Find the slope and the coordinate of the y-intercept of the given linear equation. Graph the equation using that information.

slope:

y-int:

2.3 Graph Linear Equations from Slope-Intercept Form

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Parallel lines never touch and have equal slopes (but different y-intercepts)

Perpendicular lines meet at a 90 degree angle and have slopes that are negative reciprocals.

2.3 Parallel and Perpendicular Lines

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Example:A given line has a slope of

  • What is the slope of any line parallel to this line?
  • What is the slope of any line perpendicular to this line?

Examples:Determine if the graphs of the given lines will be parallel, perpendicular or neither?

2.3 Parallel and Perpendicular Lines

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Recall from sections 3.2, 3.3 and 3.4:

2.3 Writing Linear Equations

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The Slope-Intercept Form of a linear equation in two variables is any equation in the form: y = mx + b where m is the slope of the line and (0, b) is the coordinate of the y-intercept.

Example:

What is the slope?

What is the coordinate of the y-intercept?

The Point-Slope Form of a linear equation in two variables is any equation in the form: where m is the slope of the line and is the coordinate of any point on the line.

Example:

What is the slope?

What is the coordinate of a point on the line?

2.3 Writing Linear Equations

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Example:Find the equation of the line with the given properties. Express your final answer in slope-intercept form whenever possible.

Slope is ; contains the point (3, 1)

2.3 Writing Linear Equations

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Example:Find the equation of the line with the given properties. Express your final answer in slope-intercept form whenever possible.

Line contains the points (-3, 4) and (2, 5)

2.3 Writing Linear Equations

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Example:Find the equation of the line with the given properties. Express your final answer in slope-intercept form whenever possible.

x-intercept (-4, 0) and y-intercept (0, 4)

2.3 Writing Linear Equations

slide21

Example:Find the equation of the line with the given properties. Express your final answer in slope-intercept form whenever possible.

Slope is -2 ; contains the point (0, -2)

2.3 Writing Linear Equations

slide22

Example:Find the equation of the line with the given properties. Express your final answer in slope-intercept form whenever possible.

Slope is 0 ; containing the point (3, 8)

2.3 Writing Linear Equations

slide23

Example:Find the equation of the line with the given properties. Express your final answer in slope-intercept form whenever possible.

The y-axis

2.3 Writing Linear Equations

slide24

Example:Find the equation of the line with the given properties. Express your final answer in slope-intercept form whenever possible.

Perpendicular to the line and contains the point (1, -2)

2.3 Writing Linear Equations

slide25

How to Solve a Word Problem:

  • Step 1: Read the problem until you understand it.
      • What are we asked to find?
      • What information is given?
      • What vocabulary is being used?
  • Step 2: Assign a variable to represent what you are looking for.
  • Express any remaining unknown quantities in terms of this variable.
  • Step 3: Make a list of all known facts and form an equation or inequality to solve.
  • It may help to make a labeled: diagram, table or chart, graph
  • Step 4: Solve
  • Step 5: State the solution in a complete sentence by mirroring the original question.
  • Be sure to include units when necessary.
  • Step 6: Check your result(s) in the words of the problem
      • Does your solution make sense?

2.3 Solving Applications Involving Linear Equations

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Example:Peter Griffin receives $375 per week for selling new and used cars in Quahog, Rhode Island. In addition, he receives 5% of the profit on any sales he generates. Write a linear equation that relates Peter’s weekly salary S when he has sales that generate a profit of p dollars.

2.3 Solving Applications Involving Linear Equations

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Example:Quahog Power and Light Company supplies electricity to residential customers for a monthly customer charge of $5.17 plus 10.07 cents per kilowatt-hour for up to 1000 kilowatt hours.

a. Write a linear equation that relates the monthly charge C, in dollars, to the number x of kilowatt hours used in a month

b. Graph this equation.

c. What is the monthly charge for using 200 kilowatt hours?

d. Interpret the slope of the line.

2.3 Solving Applications Involving Linear Equations

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Independent Practice

You learn math by doing math. The best way to learn math is to practice, practice, practice. The assigned homework examples provide you with an opportunity to practice. Be sure to complete every assigned problem (or more if you need additional practice). Check your answers to the odd-numbered problems in the back of the text to see whether you have correctly solved each problem; rework all problems that are incorrect.

Read pp. 173-185

Homework: pp. 185-189 #45-67 odds (give your answer in slope-intercept form only), 77-87 odds, 95, 101, 103, 105, 115, 117, 119

2.3 Lines