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3.4 Graph of Linear Equations

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Objective 1 Use the slope-intercept form of the equation of a line. Slide 3.4-3

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Use the slope-intercept form of the equation of line. In Section 3.3, we found the slope of a line by solving for y. In that form, the slope is the coefficient of x. For, example, the slope of the line with equation y = 2x + 3 is 2. So, what does 3 represent? Suppose a line has a slope m and y-intercept (0,b). We can find an equation of this line by choosing another point (x,y) on the line as shown. Then we use the slope formula. Slide 3.4-4 Rewrite. Add b to both sides. Multiply by x. Subtract in the denominator. Change in x-values Change in y-values

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Use the slope-intercept form of the equation of line. The result is the slope-intercept form of the equation of a line, because both the slope and the y-intercept of the line can be read directly from the equation. For the line with the equation y = 2x + 3, the number 3 gives the y- intercept (0,3). Slope-Intercept Form The slope-intercept form of the equation of a line with slope m and y- intercept (0,b) is Where m is the slope and b is the y-intercept (0,b). Slide 3.4-5

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Identify the slope and y-intercept of the line with each equation. Solution: Slide 3.4-6 Identifying Slopes and y-Intercepts Slope: − 1 y-intercept: (0,0) Slope: y-intercept: (0,− 6) CLASSROOM EXAMPLE 1

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Write an equation of the line with slope −1 and y-intercept (0,5). Solution: Slide 3.4-7 Writing an Equation of a Line CLASSROOM EXAMPLE 2

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Objective 2 Graph a line by using its slope and a point on the line. Slide 3.4-8

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Graph a line by using its slope and a point on the line. Graphing a Line by Using the Slope and y-Intercept Step 1: Write the equation in slope-intercept form, if necessary, by solving for y. Step 2: Identify the y-intercept. Graph the point (0,b). Step 3: Identify slope m of the line. Use the geometric interpretation of slope (“rise over run”) to find another point on the graph by counting from the y-intercept. Step 4: Join the two points with a line to obtain the graph. Slide 3.4-9

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Solution: Graph 3x – 4y = 8 by using the slope and y-intercept. Slope intercept form Slide 3.4-10 Graphing Lines by Using Slopes and y-Intercepts CLASSROOM EXAMPLE 3

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Solution: Graph the line through (2,−3) with slope Make sure when you begin counting for a second point you begin at the given point, not at the origin. Slide 3.4-11 Graphing a Line by Using the Slope and a Point CLASSROOM EXAMPLE 4

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Objective 3 Write an equation of a line by using its slope and any point on the line. Slide 3.4-12

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Write an equation of a line by using its slope and any point on the line. We can use the slope-intercept form to write the equation of a line if we know the slope and any point on the line. Slide 3.4-13

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Solution: Write an equation, in slope-intercept form, of the line having slope −2 and passing through the point (−1,4). The slope-intercept form is Slide 3.4-14 Using the Slope-Intercept Form to Write an Equation CLASSROOM EXAMPLE 5

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Write an equation of a line by using its slope and any point on the line. There is another form that can be used to write the equation of a line. To develop this form, let m represent the slope of a line and let (x 1,y 1 ) represent a given point on the line. Let (x, y) represent any other point on the line. Point-Slope Form The point-slope form of the equation of a line with slope m passing through point (x 1,y 1 ) is Slope Given point Slide 3.4-15 Multiply each side by x − x 1. Definition of slope Rewrite.

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Solution: Write an equation of the line through (5,2), with the slope Give the final answer in slope-intercept form. Slide 3.4-16 Using the Point-Slope Form to Write Equations CLASSROOM EXAMPLE 6

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Objective 4 Write an equation of a line by using two points on the line. Slide 3.4-17

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Many of the linear equations in Section 3.1−3.3 were given in the form called standard form, where A, B, and C are real numbers and A and B are not both 0. Slide 3.4-18 Write an equation of a line by using two points on the line.

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Find an equation of the line through the points (2,5) and (−1,6). Give the final answer in slope-intercept form and standard form. Solution: The same result would also be found by substituting the slope and either given point in slope-intercept form and then solving for b. Slide 3.4-19 Writing the Equation of a Line by Using Two Points Standard form Slope-intercept form CLASSROOM EXAMPLE 7

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Slide 3.4-20 Summary of the forms of linear equations.

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Objective 5 Write an equation of a line that fits a data set. Slide 3.4-21

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Solution: Use the points (3, 4645) and (7, 6185) to write an equation in slope-intercept form that approximates the data of the table. How well does this equation approximate the cost in 2005? The equation gives y = 5415 when x = 5, which is a very good approximation. Slide 3.4-22 Writing an Equation of a Line That Describes Data CLASSROOM EXAMPLE 8

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