Lesson. Objectives Find the slope of a line from the graph of the line. Find the slope of a line given two points on the line. Activity The Slope of a Line A surveyor places two stakes, A and B, on the side of a hill. Stake A is 0 feet lower than Stake B. If the horizontal distance between the stakes is 00 feet, what is the slope of the hill? The y-axis Graph these three points on the same Cartesian coordinate system: (0, ), (0, 0), (0, ). Describe the location of the three points. All three points lie on the y-axis (the vertical axis). The x-value of each point is zero. However, the y-values are different. Any ordered pair that has an x-value equal to zero must identify a point somewhere along the y-axis. Thus, the equation x 0 describes the y-axis. Activity The x-axis Graph these three points on the same Cartesian coordinate system: (, 0), (0, 0), (, 0). Describe the location of the three points. What is true about the y-value for each of these points? What is the equation that describes the x-axis? Activity Equal Coordinates Graph these three points on the same Cartesian coordinate system: (, ), (0, 0), (, ). Describe the location of the three points. Do all of these points lie on one straight line? What is true of the x-values and y-values of all these points? Write an equation that describes this line.. The Slope of a Line 07
In the preceeding Activities, you have used equations to describe three lines. You can also graph the lines on the same coordinate system. y-axis y = x 6 (x = 0) (y = 0) x-axis 6 0 6 6 Slope Imagine that each of the three equations (x 0, y 0, and y x) represents a hill. You have to climb each hill. Which hill is easiest to walk up? The y 0 hill (or x-axis) is the easiest it has no rise at all. You can say that the line y 0 has a steepness of zero. The steepness of a line is called the slope. Thus, the line y 0 has a slope of zero. Which equation represents a hill so steep that it is impossible to climb? You cannot walk up the x 0 hill (or y-axis) at all. The slope of this hill is so great that you cannot assign a number to it. The slope of the line x 0 is undefined. Which equation represents a hill that is fairly steep, but one you could still climb? The y x hill has a slope somewhere between the x-axis (with a slope of zero) and the y-axis (with a slope that is undefined). How can you find the slope of the line y x? Rise Run The slope of a line is a measure of its steepness or tilt. The steepness of a line (or a hill) is found by comparing its vertical rise to its horizontal run. A very steep hill has a large amount of vertical rise for the given amount of horizontal run shown. 08 Chapter Linear Equations
A road with a gentle slope has a small amount of vertical rise for the same amount of horizontal run. Run Rise Slope The slope of a line is the ratio of the distance of the rise to the distance of the run, where the distances are measured with the same units. y Run is to the right, so the slope is positive. Run = B Rise = 6 D Run Rise A C x Lines have slopes that are positive, negative, zero, or undefined. To find the slope of a line, you need two points on the line. Imagine yourself walking from point A to point B. However, in this imaginary walk, you must first move up (or down), and then go right or left; you cannot go diagonally. Count the steps (or units) going up to find the rise. Then count the steps (or units) either left or right (the run) to reach B. If you move to the right, the number for the run is positive; if you move to the left, the number for the run is negative. Thus, slope is a rate of change. Once you know the value of the rise and run, write a fraction with the rise as the numerator and the run as the denominator. This fraction representing the ratio rise run is the slope of the line. The slope of line AB is 6 or.. The Slope of a Line 09
Critical Thinking Why is the slope of the line passing through C and D negative? Example Critical Thinking Finding Slope Critical Why is Thinking the slope of Why the line is the passing slope through of the line C passing through Refer and to the Dopening negative? paragraph From and CDto in negative? D, this the rise lesson is From positive, about C to the and D, the two the rise run stakes is positive, negative. the surveyor and the run is negative. placed. What is the slope of the hill? EXAMPLE Finding EXAMPLE Slope Finding Slope Solution A surveyor places two A surveyor stakes, Aplaces and B, two on the stakes, side Aof and a hill. B, on Stake the Draw Aside a of d a hill. Stake A 0 Draw is a diagram. 0 feet lower The rise than is Stake 0 feet. B. lower If The the run horizontal than is Stake 00 feet. distance B. If The the slope between horizontal is the distance 00 or 0 between th or 0.. stakes is 00 feet, what stakes is the is 00 slope feet, of what the hill? is the slope of the hill? SOLUTION SOLUTION Draw a diagram. The Draw rise is a diagram. 0 feet. The The run rise is is 00 feet. The run slope is is 00 feet. The slope 0 0 Stake B 00 or 0 or 0.. 00 or 0 or 0.. Stake A 0 ft Activity 00 ft Stake B Discovering the Slope Formula Stake A Stake A 0 ft 0 Find the slope of AC. How many units did you rise? How many 00 ft 00 ft units did you run? ACTIVITY Discovering ACTIVITY the Discovering Slope Formula the Slope Formula Use the y-elements of the Find ordered the pairs slope for of points AC. Find How the A and slope many C. of units AC. did How you many rise? units Howdid you rise? How many Look for units a relationship did you many run? with units ;rise the 8 did units; you run run? 6 units ;rise 8 units; run 6 units rise units named in Step. Use the y-elements Use of the y-elements ordered pairs of for the points ordered A and pairs C, for points A and C, Do look the for x-elements a relationship look of the with for a the relationship rise units with named the in rise Step units. named in Step. the difference between ordered pairs for points the and difference is 8. A and between C and is 8. Does the x-elements have the same relationship Does of the ordered x-elements pairs with of for the points ordered A and pairs C for points A and C have the same relationship the run units named have in the with Step same the? relationship run units named with the in run Step units? named in Step yes; the difference between yes; the and difference is 6. between and is 6. Explain why a rise Explain why rise Explain of 8 units of units why and and a rise run run of of 86 units equals and equals run a of slope slope 6 units equals a slope of of. 8 6 reduces to. of. 8 6 reduces to. 8 6 reduces to Repeat Steps Repeat Steps Repeat for CB. for CB. ; Steps ; rise rise 6 units; 6 units; for CB. run run ; units; units; rise yes; yes; 6 the the units; difference run units; yes; the difference is 6;Yes; the difference difference is is 6;Yes; is ; yes; 6 the difference is ; 6 reduces to. reduces to. 66 Repeat Steps 6 Repeat for AB. AB. Steps ;rise 9; units; for run AB. 9 ;rise units; 9 yes; units; the difference run 9 units; is ; yes; the difference yes; the difference is ; yes; is 9; the because difference difference the is ; is rise 9; and yes; since run the the are difference rise both and negative is run 9; are since both the slope is the rise and run are both 7 Generalize negative the slope how is to positive. negative find the the slope is of positive. a line using the ordered pairs. 7 Generalize the 7how Generalize to find the the slope how of to a find line the using slope theof a line using the ordered pairs. Subtract ordered the y elements pairs. Subtract for the numerator the y elements and subtract for the numerator the and subtract the x elements for the denominator. 0 Chapter Linear Equations x elements for the denominator. Stake
The Slope Formula The coordinates of any two points on a line determine its slope. The difference between the y-coordinates is the rise. The difference between the x-coordinates is the run. This gives a formula for finding slope. Because the slope is the ratio rise run, the slope can also be written in the following way slope difference of y-coordinates difference of x-coordinates. To find the slope of a line between two points, use the slope formula. Slope Formula If A(x, y ) and B(x, y ) are two points on line AB, then the slope of AB. x x y y When you use the slope formula to find the slope of a line between two points, be sure to subtract the coordinates in the same order. Example The Slope Between Two Points Find the slope of the line that contains A(, ) and B(, ). Solution Method Make a sketch. Start at the lower point, A. Move up to find a rise of 7 units. To reach point B, move to the right units. This is a run is i of. Thus, the slope is 7. y 6 run is B (, ) rise is 7 6 0 6 (, ) A 6 x Method Use the slope formula. y y Slope of AB x ( ) 7 x ( ) The slope is the ratio 7.. The Slope of a Line
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7. It is 0 miles from Johnson City to Putnam. The elevation of Johnson City is,000 feet. The elevation of Putnam is,00 feet. What is the average rate of increase in elevation per mile from Johnson City to Putnam? 8. What is the positive slope of the roof at the right? 8 ft 6 ft 9. In a landing approach, an airplane maintains a constant rate of descent of 0 feet for every 00 feet traveled horizontally. What is the positive slope of the line that represents the landing approach of the plane? 0 Without graphing, determine if each set of points lie on the same line. 0. (, ), (, ), (, ). (7, 7), (, ), (, ). (8, ), (6, ), (, 6), (, 7). (0, 0), (, ), (, ), (0, ) Mixed Review For each situation, write and solve an equation.. The amount of water flowing over a dam at noon is. million gallons per hour more than its rate at mid-morning. When the water flow was tested at noon, it had reached 8 million gallons per hour. What was the rate of the water flow at mid-morning?. Keshia sells her inventory for twice what she pays. After expenses of $0 are deducted, Keshia finds she has $680 left. What did Keshia pay for her initial inventory? 6. Ramon is carpeting a rectangular room with a perimeter of 0 feet. One side of the room is feet longer than the other. Find the length of the longer side. Solve each equation. Check your answer. 7. d () 8. x% of 0 9. 0 6 r. The Slope of a Line