7.3. Slope-Point Form. Investigate Equations in Slope-Point Form. 370 MHR Chapter 7

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1 7. Slope-Point Form Focus on writing the equation of a line from its slope and a point on the line converting equations among the various forms writing the equation of a line from two points on the line solving problems involving equations in slope-point form You can measure the length of an arena with a ruler, use scissors to cut our grass, or loosen a screw with a paperclip. All of these are possible, but are the using the best tool for the job? In mathematics and in life, using an inappropriate tool ma prevent ou from completing our task or make it take longer to finish. You have eplored tools for writing linear relations in two forms. This section introduces a third form, slope-point form. Each form is best suited to certain situations. Materials grid paper ruler Investigate Equations in Slope-Point Form. Square ABCD in Figure is a composite of four different polgons. The lengths of the sides are shown. What is the area of square ABCD? B 5 C 5 5 A 8 D Figure 7 MHR Chapter 7

2 . Square ABCD is reassembled to form rectangle EFGH, shown in Figure. What is the area of rectangle EFGH? F 5 8 G 5 5 E 8 5 H Figure. There is a discrepanc between the areas of the quadrilaterals shown in Figures and. How is Figure deceiving? Justif our answer.. On grid paper, draw a line that does not pass through the origin. Label points J, K, and L on the line. Determine the slope of our line. a Determine the equation of our line using point J and the slope-intercept form, = m + b. b Use points K and L to determine equations of our line. Compare our equations. 5. Let P(, represent a point on a line. Develop an equation of the line with slope m using point P. 6. Work with a partner. Have our partner test the equation ou developed using his or her line from step. If ou know one point on a line, how can ou use the slope to determine a second point? 7. Reflect and Respond Describe how to determine the equation of a line using the slope and a point on the line. 8. Show how the slope-point form of a linear equation can be developed b using the slope formula, m = What tpe of line cannot be written in slope-point form? Wh?. Is the following statement alwas true, sometimes true, or never true? Eplain. To determine the equation of a nonvertical line in slope-point form, ou can use the coordinates of an point on the line. slope-point form the equation of a nonvertical line in the form -, where m is the slope and (, are the coordinates of a point on the line 7. Slope-Point Form MHR 7

3 Link the Ideas _ The slope of a non-vertical line can be determined using m = Δ Δ. If (, is one point on the line, then (, could represent an other point on the line. Substitute the coordinates of these two points into the slope formula, m = - -. The slope of the line could be written as m = - -. (, (, - - Multipling both sides of the above equation b ( - gives ( - m = ( - ( - - ( - m = ( - ( - - m( - = - This equation is called the slope-point form of a non-vertical line through point (, with slope m. The slope-point form is commonl written as -. Eample Write the Equation of a Line Using a Point and the Slope a Use slope-point form to write an equation of the line through (-, 5 with slope -. b Epress the equation in slope-intercept form, = m + b. c Graph the linear relation using technolog. Solution a Substitute - for m and the coordinates of the point (-, 5 for (,. - - (5 = -( - (- - 5 = -( + The equation in slope-point form is - 5 = -( +. 7 MHR Chapter 7

4 b To epress the equation in slope-intercept form, isolate. - 5 = -( + = -( = = - - In slope-intercept form, the equation is = - -. c What strategies could ou use to sketch the graph? The equation = - - is written in the form = m + b. So, the slope is -. This is consistent with the value given in the question. Your Turn a Use slope-point form to write an equation of the line through (, with slope. Sketch a graph of the line. b Epress the equation in slope-intercept form, = m + b. Sketch a graph of this line. c Compare our graphs. Eample Determine the Equation of a Line Using Two Points a Use slope-point form to write an equation of the line through (, and (5, -. b Sketch a graph of the line. c Rewrite the equation in general form, A + B + C =. Solution a Points on the line are given. So, ou need to determine the slope. Use the two given points, (, and (5, -. m = ( m = 5 - m = m = _ 7. Slope-Point Form MHR 7

5 In slope-point form, substitute _ for m and the coordinates of either point (, or (5, - for (,. Using (, for (,, Using (5, - for (,, ( = _ ( - - (- = _ ( = _ Both + = _ ( - + = _ ( - 5 ( - and + = _ How can ou verif that these equations are equivalent? ( - 5 are slope-point forms of the equation of the line through (, and (5, -. b (5, - (, c Epress + = _ general form. ( - in + = _ ( - ( = ( = - 9 = = ( + = ( _ Epress + = _ ( - 5 in general form. + = _ ( - 5 ( + = ( _ ( - 5 ( + = ( = - 5 = The equation, in general form, for the line through (, and (5, - is =. Your Turn Use slope-point form to write an equation of the line through (-5, and (-,. Eplain our steps. Then, write the equation in general form, A + B + C =. 7 MHR Chapter 7

6 Eample Model a Real-Life Situation Brad Zdanivsk is enthusiastic about mountain climbing. He is a quadriplegic and used custom gear as he climbed the Stawamus Chief in Squamish, BC, on Jul, 5. Supposed he moved at a constant rate and climbed the 66-m summit in pitches (sections. Each pitch was approimatel 6 m in height. At 5:5 a.m., Brad started his climb 6 m below the top of his first pitch. B 5:55 a.m., he was m below the top of the first pitch. a Write an equation that describes Brad s distance, d, in metres, below the top of the first pitch in terms of t minutes past 5:5 a.m. Epress the equation in = m + b form. b How long did it take Brad to reach the top of the first pitch? c In total, Brad spent 8.5 h changing ropes between pitches. How long did it take Brad to climb the Stawamus Chief? Web Link To learn more about mountain climbing, go to and follow the links. Solution a Brad was 6 m below the top of his first pitch at min past 5:5 a.m. After min, he was m below the top of his first pitch. As coordinate pairs (t, d, the data ma be represented as (, 6 and (,. Use these points to determine the slope of the line. m = d - d t - t m = m = m = - _ m = - m = - Brad s distance to the top of the pitch was decreasing at a rate of m/min. Substitute the slope, -, and the coordinates of either point (, 6 or (, into the slope-point form of an equation. Using point (, 6, d - d = m(t - t d - 6 = -(t - d - 6 = -t d = -t + 6 How could ou verif our equation? In slope-intercept form, the equation d = -t + 6 represents Brad s distance below the first pitch. 7. Slope-Point Form MHR 75

7 b At the top of the first pitch, d =. Determine t. d = -t + 6 = -t + 6 t = 6 t = It took Brad min or.5 h to reach the top of the first pitch. c To climb the pitches, it took Brad (.5 h = 5.5 h. Adding 8.5 h to change ropes, it took Brad h to climb the Stawamus Chief. Your Turn A famil drives at a constant speed Wrigle from Wrigle, NT, to visit relatives in Fort Providence, NT. When the start driving at 9: a.m., the are 5 km from Fort Fort Simpson Providence. At : p.m., the reach Fort Fort Providence Simpson, located 5 km from Fort Providence. a Write an equation that describes their distance, d, in kilometres, from Fort Providence in terms of t hours past 9: a.m. b What time will the famil reach Fort Providence? Ke Ideas For a non-vertical line through the point (, with slope m, the equation of the line can be written in slope-point form as -. A line through (-, 5 has a slope of. The slope-point form of the equation of this line is = ( - (- - 5 = ( + (-, An equation written in slope-point form can be converted to either slope-intercept form or general form. An point on a line can be used when determining the equation of the line in slope-point form. 76 MHR Chapter 7

8 Check Your Understanding Practise. Rewrite each equation from slope-point form to slope-intercept form, = m + b, and general form, A + B + C =. a + = - 5 b + = ( + c - 6 = ( + d + = -5( + e - = - _ ( + 8 f + 9 = - _ ( - 6. Write an equation in slope-point form, -, of each line passing through the given point. a b (, (, - c d -6 (, (, - -. Determine the equation of each line using slope-point form. Then, epress each equation in slope-intercept form and in general form. a (5, -, m = 6 b (-, -5, m = - c (-8,, m = _. Consider the line represented b - = _ d (, -6, m = - _ ( - 6. a Identif the slope and a point on the line. b Eplain how ou could sketch the graph of the line using the slope and a point on the line. 7. Slope-Point Form MHR 77

9 5. Write an equation in slope-point form, -, of the line passing through the given points. a (, 5 b (-, (, (-, - c d (, (-, (, (, - 6. Use slope-point form to write an equation of a line through each pair of points. Epress each equation in the form = m + b and in the form A + B + C =. a (5, and (, -7 b (5, -8 and (, c (, 5 and (, 6 d (8, - and (, -6 e (5, - and (, f (, 6 and (-, 7. Terr s teacher writes the following on the board: The four equations listed represent onl two different lines. Which equations represent the same line? - = ( + - = ( = ( + - = ( - a Describe possible strategies students could use to answer the question. b Which equations represent the same line? Justif our answers. 78 MHR Chapter 7

10 Appl 8. Identif the slope and a point on each line. Sketch a graph of each line. Use graphing technolog to check our graphs. a - = ( - b + = - ( + d + 6 = - _ ( - 5 c - = _ 9. Consider the line passing through the points (, and (-, 6. a Work with a partner to develop at least two different strategies for determining the -intercept of the line. b What is the -intercept of the line?. A line passes through (, and (, 7. a Using onl slope-intercept form, = m + b, write the equation of this line. b Determine the equation of the line using onl slope-point form. c Compare the two equations graphicall.. Write the equation of each line using slope-point form. Then, convert to slope-intercept form. a slope of and through (, -5 b same slope as + = 5 and through (-, c same slope as the line = and the same -intercept as the line - = d same -intercept as + = 8 and through (,. What is the equation of each line in slope-point form? Convert each equation to general form. a slope of and -intercept of b same slope as = + 5 and through (, - c same -intercept as the line + = and through (, d -intercept of and -intercept of -6. An iron horse pumpjack starts to pump crude oil into a tank at a constant rate of. m /h. After h, the tank contains 9 m of oil. a Write an equation that describes the volume, V, in cubic metres, of oil in the tank after t hours. b The tank can hold a maimum of 55 m of oil. How long will it take to fill the tank? c Was the tank empt before it started filling? Eplain. Did You Know? Oil pumpjacks are common in western Canada. The are a traditional method of oil recover. The surface deposits of the Athabasca Oil Sands in present da northern Alberta were once used b the Cree and Dene peoples to waterproof their canoes and other items. 7. Slope-Point Form MHR 79

11 . The graph shows the linear V Sound Velocit relationship between the 5 velocit of sound, V, in metres per second, and the temperature, t, in degrees Celsius, of dr air. At 6 C, the 5 velocit of sound is 5 m/s. (6, 5 At 6 C, it is m/s. a What is the slope of the line? b What rate of change does the 5 5 slope represent? Temperature ( C c What is the equation of the line? d Determine the velocit of sound at 5 C. e What is the air temperature when the velocit of sound is 8 m/s? 5. What is the -intercept of a line with a slope of _ and an -intercept of? Velocit (m/s (6, t 6. Determine the -intercept of a line through (, having a -intercept of. Web Link To learn more about the components of population growth in parts of Canada, go to and follow the links. 7. Suppose Canada s population has grown steadil since. In, the population was. million. In 9, it was.7 million. a Let t represent the number of ears since. Let p represent the population of Canada in millions. Write the coordinates of two points in the form (t, p. b Determine the slope of the line through the points. c What rate does the slope represent? d Write an equation to represent population growth in Canada since. e Predict Canada s population in 7. 8 MHR Chapter 7

12 8. Suppose our friend s dinner tonight consists of one steak and mini potatoes. The steak has approimatel g of protein. The nutrition facts label shows the number of grams of protein per number of mini potatoes. a Write an equation relating the protein, p, in the meal to the number of potatoes, n, eaten. Use the data in the nutrition facts label. b What is the slope of the line? What does the slope represent? c What is the p-intercept of the line? Wh is the p-intercept not zero? d Suggest a reasonable domain and range for the graph. Did You Know? In Canada, potato production is a multimillion-dollar industr. Manitoba produces the second greatest amount of potatoes in Canada. Etend 9. Write the equation of a line with an -intercept of n and a slope of m.. A line passes through the point of intersection of the lines = - _ - 6 and = +. Determine the equation of the line if it has a slope of _. 7. Slope-Point Form MHR 8

13 Create Connections. How can ou develop the slope-intercept form, = m + b, b substituting a point into -?. To determine the equation of a line in slope-point form, ou need to know two pieces of information about the line. List three sets of information that would allow ou to determine the equation of a line.. To solve a particular problem ou ma want to write a linear equation in one of the three forms. You ma wish to use slopeintercept form, = m + b; general form, A + B + C = ; or slope-point form, -. Create a visual that helps ou decide which form ou should start with. Materials SI measuring tape grid paper ruler. MINI LAB Unit Project Paleontologists can predict the anatom of humans and animals based on skeletal remains. Step Step Step Step Work with a partner of the same gender as ourself. Measure and record the length of each other s humerus bone. It runs from the shoulder to the elbow. Measure and record each other s height without shoes. Collect and share our data with other students of the same gender. Record all data. Use grid paper to plot the data as coordinate pairs. Label the aes and scale used. Draw a straight line that represents the data. What is the equation of this line? Measure the humerus bone of a teacher of the same gender as ou. Use our equation to predict the height of the teacher. Compare the teacher s actual height with our predicted height. 8 MHR Chapter 7