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Section_1.10-Lines

# Section_1.10-Lines - Section 1.10 Lines The Slope of a Line...

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Section 1.10 Lines The Slope of a Line EXAMPLE: Find the slope of the line that passes through the points P (2 , 1) and Q (8 , 5). Solution: We have m = y 2 - y 1 x 2 - x 1 = 5 - 1 8 - 2 = 4 6 = 2 3 1

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EXAMPLE: Find the slope of the line that passes through the points P ( - 2 , - 1) and Q ( - 8 , 5). Solution: We have m = y 2 - y 1 x 2 - x 1 = 5 - ( - 1) - 8 - ( - 2) = 6 - 6 = - 1 EXAMPLE: Find the slope of the line that passes through the points P ( - 3 , 1) and Q (5 , 6). Solution: We have m = y 2 - y 1 x 2 - x 1 = 6 - 1 5 - ( - 3) = 5 8 Point-Slope Form of the Equation of a Line From the picture above it follows that y - y 1 x - x 1 = m which can be rewritten as y - y 1 = m ( x - x 1 ) EXAMPLE: Find an equation of the line through (1 , - 3) with slope - 1 2 and sketch the line. 2
EXAMPLE: Find an equation of the line through (1 , - 3) with slope - 1 2 and sketch the line. Solution: Using the point-slope form with m = - 1 2 , x 1 = 1 , and y 1 = - 3 , we obtain an equation of the line as y - ( - 3) = - 1 2 ( x - 1) y + 3 = - 1 2 ( x - 1) 2 y + 6 = - x + 1 x + 2 y + 5 = 0 EXAMPLE: Find an equation of the line through the points ( - 1 , 2) and (3 , - 4) . Solution: The slope of the line is m = - 4 - 2 3 - ( - 1) = - 6 4 = - 3 2 Using the point-slope form with x 1 = - 1 and y 1 = 2 , we obtain y - 2 = - 3 2 ( x - ( - 1)) y - 2 = - 3 2 ( x + 1) 2 y - 4 = - 3 x - 3 3 x + 2 y - 1 = 0 3

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Slope-Intercept Form of the Equation of a Line From the picture above it follows that y - b = m ( x - 0) which can be rewritten as y = mx + b EXAMPLE: Find an equation of the line with slope 3 and y -intercept - 2 . Solution: Since m = 3 and b = - 2 , from the slope-intercept form of the equation of a line we get y = 3 x - 2 EXAMPLE: Find an equation for the line that has x -intercept 6 and y -intercept 4. Solution: Since the y -intercept is 4, it follows that b = 4 . Since the x -intercept is 6, it follows that 0 = 6 m + b Plugging in 4 into this equation, we get m = - 2 3 . Therefore an equation for the line that has x -intercept 6 and y -intercept 4 is y = - 2 3 x + 4 EXAMPLE: Find the slope and y -intercept of the line 3 y - 2 x = 1 .
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