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Section 1.3 • Linear Equations in Two Variables 9 Name______________________________________________ Section 1.3 Linear Equations in Two Variables Objective: In this lesson you learned how to find and use the slopes of lines to write and graph linear equations in two variables. Important Vocabulary Define each term or concept. Slope The number of units a nonvertical line rises (or falls) vertically for each unit of horizontal change from left to right. Parallel Two distinct nonvertical lines are parallel if and only if their slopes are equal. That is, m1 = m2. Perpendicular Two nonvertical lines are perpendicular if and only if their slopes are negative reciprocals of each other. That is, m1 = − 1/m2. I. Using Slope (Pages 25−26) What you should learn How to use slope to graph linear equations in two variables The equation y = mx + b is called a linear equation in two variables because . . . the graph of this equation in the two variables x and y is a straight line. A line whose slope is positive rises from left to right. A line whose slope is negative falls from left to right. The slope-intercept form of the equation of a line is y = mx + b , where m is the slope and the y-intercept is ( 0 , b ). A vertical line has an equation of the form x=a . The equation of a vertical line cannot be written in the form y = mx + b because . . . the slope of a vertical line is y 5 undefined. Example 1: Explain how to graph the linear equation y = − 2/3x − 4. Then sketch its graph. Because b = − 4, the y-intercept is (0, − 4). Because the slope is − 2/3, the line falls 2 units for every 3 units the line moves to the right. 3 1 x -5 -3 -1 -1 -3 -5 Larson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide IAE Copyright © Houghton Mifflin Company. All rights reserved. 1 3 5 10 • Chapter 1 Functions and Their Graphs Example 2: Sketch and label the graph of (a) y = − 1 and (b) x = 3. y y (a) (b) 5 5 -5 -3 3 3 1 1 -1-1 1 3 5 x -5 -3 -1-1 -3 -3 -5 -5 1 3 5 x II. Finding the Slope of a Line (Pages 27−28) The formula for the slope of a line passing through the points (x1, y1) and (x2, y2) is m = (y2 − y1)/(x2 − x1) What you should learn How to find slopes of lines . To find the slope of the line through the points (− 2, 5) and (4, − 3), . . . subtract − 3 from 5 and divide this result by the difference of − 2 and 4. If a line falls from left to right, it has line is horizontal, it has vertical, it has right, it has undefined positive zero negative slope. If a slope. If a line is slope. If a line rises from left to slope. III. Writing Linear Equations in Two Variables (Page 29) The point-slope form of the equation of a line is y − y1 = m(x − x1) . The point-slope form is most useful for . . . What you should learn How to write linear equations in two variables finding the equation of a line. The two-point form of the equation of a line is y − y1 = (y2 − y1)/(x2 − x1)⋅(x − x1) . Larson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide IAE Copyright © Houghton Mifflin Company. All rights reserved. Section 1.3 • Linear Equations in Two Variables Name______________________________________________ Example 3: Find an equation of the line that passes through the points (1, 5) and (− 3, 7) using (a) the point-slope form and (b) the slope-intercept form. (a) y − 5 = − 0.5(x − 1) (b) y = − 0.5x + 5.5 IV. Parallel and Perpendicular Lines (Page 30) What you should learn How to use slope to identify parallel and perpendicular lines The relationship between the slopes of two lines that are parallel is . . . that the slopes are equal. The relationship between the slopes of two lines that are perpendicular is . . . that the slopes are negative reciprocals of each other. A line that is parallel to a line whose slope is 2 has slope 2 . A line that is perpendicular to a line whose slope is 2 has slope − 1/2 . V. Applications of Slope (Pages 31−33) In real-life problems, the slope of a line can be interpreted as either a(n) ratio or a(n) rate . If the x-axis and y-axis have the same unit of measure, then the slope has no units and is a ratio . If the x-axis and y-axis have different units of measure, then the slope is a change rate or rate of . Describe a real-life situation in which slope is a ratio. Answers will vary. Describe a real-life situation in which slope is a rate of change. Answers will vary. Linear or straight-line depreciation is . . . used when the same amount of depreciation is claimed each year over the useful life of property. Larson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide IAE Copyright © Houghton Mifflin Company. All rights reserved. What you should learn How to use linear equations in two variables to model and solve real-life problems 11 12 Chapter 1 • Functions and Their Graphs The prediction method in which an estimated point does not lie between the given points is called . linear extrapolation When the estimated point lies between two given points, the procedure is called linear interpolation . Every line has an equation that can be written in general form, which is given as , where A Ax + By + C = 0 and B are not both zero. Additional notes y y 5 5 3 3 3 1 -5 -3 y 5 -1-1 1 1 3 5 x -5 -3 -1-1 1 1 3 5 x -5 -3 -1-1 -3 -3 -3 -5 -5 -5 1 3 5 x Homework Assignment Page(s) Exercises Larson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide IAE Copyright © Houghton Mifflin Company. All rights reserved.