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4i UNIT . .:_ .:,..":. & . :.....: tl 5 ' .-,--, :r, a-:\ -_ i. =c'l ".t i't-/ ;:!- -rL i'-'" g i. {{ _1 i{ i 'i -; '..-'- ,a . ," . -l il ri-. :i J:,r Ii :l'_ - -. : ].,..','""".;....-..,,.1ix.: ". -'-1 -.") .i ':ii:;i;:,;;1:;;;;:, 1 .1:;;:|ii!:;:;:i;ilW;,i?' illliLti:iiiilit*;#j;t:lli;iliiii;!:l{;,: il:it:li:ti;iliL,.: .t .",.,,,,,: 292 uN tr s Graphs and Linear Equations List of Objectives To graph points on a rectangular coordinate system To determine a solution of a linear equation in two variables To solve application problems To graph an equation of the forry 1 - To graph an equation of the forrn Ax mx * b * B! - C To find the x- and yintercepts of a straight line To find the slope of a straight line To graph a line using the slope and 7-intercept To find the equation of a line using the equation) - mx * To find the equation of a line using the point-slope formula b UNIT I SECTION 1 1.1 Objective 293 Graphs and Linear Equations The Rectangular Coordinate System To graph points on a rectangular coordinate system A rectangular coordinate system ll Quadrant i. ..i.. -i .'.. is ... i formed by two number lines, one horizontal and one vellical, that intersect ilir at the zero point of each line. The point of intersection is called the origin. The two lines are called the coor- vertical hoiizoniaii ''''''t . i !axtsi dinate axes, or simply axes. EAD' The axes determine a plane and divide the plane into four regions, called quadranls. The quadrants are numbered counterclockwise from I to lV. axis t : tzJ4 Ouadrant Each point in the plane can be identified by a pair of numbers called an ordered pair, The first number of the pair measures a horizontal distance and is called the ab- scissa. The second number of the pair measures a vertical distance and is called the ordinate. The coordinates of a point are the numbers in the ordered pair associated with the point. o O r t- horizontal distance o o 1 ordered pair --> (3 j E uo ! C 6 , vertical distance 4) t_ ordinate "ba",ar" The graph of an ordered pair is a point in the plane. The graphs of the points (2,3) and (3,2) are shown at the right. Notice that they are different points, The order in which the numbers in an ordered pair appear is c o O -J(2,3) -i-1(3, 2) important. Example 1 Graph the ordered pairs (-2,-3), (3,-2), (1,3), and Example 2 ( - 1 ,3), (1 ,4), (-4,0), (-2,-1) (4,1). Solution Graph the ordered pairs and Your solution ' i : -;1 € N \t ci o o 6 at 294 UN Example 3 lT 8 Graphs and Linear Equations Find the coordinates of each of Example 4 the points. -+-3-2-t .Ai i o i- Find the coordinates of each of the points. 34 ....i.....t. oD: :: t: Solution A (-4,-2) Your solution B (4,4) c (0, n /? ? \v, Example 5 a) b) -3) 2\ Example 6 Name the abscissas of points A and C. Name the ordinates of points B and D. a) b) Name the abscissas of points A and C. Name the ordinates of points B and D. oo I o @ o @ E o I ! d -4 E o c o o Solution a) b) Abscissa Abscissa Ordinate Ordinate of point A: of point C: of point 8: of point D: a o l\ Your solution tt ci 0 E a o a c o 0 6 o 1.2 Objective To determine a solution of a linear equation in two variables An equation of the form y-mx +b,where mandb are constants, is a linear equation in two variables. Examples of linear equations are shown at the right. Note y-3x+4 y-2x-3 Y--!x+t y--2x y-x+2 (m-3, b-4) (m=2, b=-3) (m:-!,0=t1 (m=-2,b=0) (m =1, b:2) that the exponent of each vari- able is always one. A solution of an equation in two variables is an ordered pair of numbers (x,y) which makes the equation a true staiement. UN IT 8 295 Graphs and Linear Equations ls (1,-2; a solution oI y Replace x with Replace y with - 3x - f=3x-S 5? l, the abscissa. -2,Ihe ordinate. E 5 Yes, (1 , -2) is a solution of the equation y - 3x - 5 Compare the results. lf the results are equal, the given ordered pair is a solution. lf the results are not equal, the given ordered pair is not a solution. Besides the ordered pair (1,-2), there are many other ordered pair solutions of the 5. For example, the method used above can be used to show lhat equation y = 3x - (2,1), (-1,-8), (3 -t) and (0,-5) are also solutions. ln general, a linear equation in two variables has an infinite number of solutions. By choosing any value for x and substituting that value into the linear equation, a corresponding value of y can be {ound. oo Find the ordered pair solution of y Substitute 1 lor x, Solve for y. -, - 2x - 5 corresponding to x = 1. y-2x-5 -2.1-5 _q The ordered pair sorution is I o @ o (1 . e -3) o o E o L oc c o c o o Example 7 Solution ls Example (-3,2) a solution of y-2x+2? y-2x+2 212(-3)+z -6+2 8 ls (2, -4) a solution of r- -!x-sz Your solution -4 2+-4 Example 9 No, (-3,2) is not a solution y,, o,, -tt L, o L^ - of Find the ordered pair solution ot y'3 - ?^ Example 10 1 corresponding - of Y-5x_a I | I € -1 The ordered pair solution (3,1) - -I, * 1 correspond- Your solution r --2rat 5\e./ - y ingtox=4. tox=3. Solution Find the ordered pair solution is t tt o 6 c o = oo UNIT 296 1.3 Obiective Graphs and Linear Equations 8 To solve application problems A rectangular coordinate system is frequently used in business, science, and mathematics to show a relationship between two variables. One variable is represented along the horizontal axis and the other variable is represented along the vertical axis' For example, a physics student measured the sPeed of imPact of a solid ball dropped from various heights above the ground. The results were recorded as the following ordered pairs: (1,8), (4,16), (9,24). The abscissa of an ordered Pair is the height in feet and the ordinate of an ordered pair is the speed in feet per second. For examPle, the ordered pair (4,16) corresponds to dropping the ball from a height of 4 ft and recording the speed as 16 ftls. The results are graphed at Ec o o 25 aB o. zv qq) a"z q) o 15 10 ( c 0 2 34567BI 10 Height (in feet) the right. oo I Example 11 Strategy Solution A chemist measured the temperature of a chemical reaction at different times and recorded the results as the following ordered pairs: (5,10), (10,15), and (20,25). The abscissa of an ordered pair is time measured in minutes, and the ordinate of an ordered pair is temperature measured in degrees. Graph the ordered pairs on the rectangular coordinate system. Graph the ordered Pairs (5,10), (10,15), and (20,25). (Doe .r ta f (!) ^^ E9 t' b s', 15 93 E- o.= to ',t)' . ; o(10, 15) o (5, 10)r : 5 o 12 @ A biochemist recorded the number of bacteria in a culture at different times as the following ordered pairs: (1,8), (3,10), and (4,11). The abscissa of an ordered Pair is the time in hours, and the ordinate of an ordered Pair is the number of bacteria. GraPh the ordered pairs on the rectangular coordi- o o d E IE C d c @ c nate system. Your strategy Your solution :l:: (3oi Example 5 10152025 Time (in minutes) (! q) o (! o o o _o E f z o !t !i Time (in hours) E o c I t o ah UN lT 8 297 Graphs and Linear Equations 1.1 Exercises 1. Graph the ordered pairs (-2,1), (3,-5), (-2,4), and (0,3). Graph the ordered pairs (0,-5), (-3,0), and (0,2). (0,0), Graph the ordbred pairs (-1,4), (0,2), and (4,0). (-2,-3), 2. Graph the ordered pairs (5,-1), (-3,-3), (-1,0), and (1,-1). Graph the ordered pairs (-4,5), (-3,1), (3,-4), and (5,0). Graph the ordered pairs (5,2), (-4,-1), (0,0), and (0,3). -1 t:2:0 l l , :4-i'z i t ) 298 UNIT 7. 8 Graphs and Linear Equations Find the coordinates of each of the 8. Points. 9. Find the coordinates of each of the 10. Find the coordinates of each of the points. Find the coordinates of each of the points. points. ()o x o @ o @ d E o L oc 6 c o a o o 11. a) b) Name the abscissas of points A and C. Name the ordinates of points B and D. 12. a) b) Name the abscissas of points A and C. Name the ordinates of points B and D, UN lT 8 299 Graphs and Linear Equations 1.2 Exercises 13. ls (3,4) a solution of y- -x + 7? 15. ls (-1,2) a solution 17. y 14. ls(2,-3)asolution oIy - x + 5? 16. (1,-3) a -2x - 1? ls y:lx_1? ls (4,1) a solution ot - f,x + 1? 18. ls (-5,3) a Y 19. ls (0,4) a solution oI y - |x + +? solution solution - -!x + 1? ls (-2,0) y= -+x - a solution 1? o O I o o E 21. ls (0,0) a solution ot y - 3x + 2? ls (0,0) a solution oI Y : -1*? E uo ! c E o o O Find the ordered pair solution y-3x-2 x-3. corresponding of to I Find the ordered pair solution of y= f,x X: 24, Find the ordered pair solution of y - 4x I 1 corresponding to -1 corresponding to y: ]x - 2 corresponding to X=4. 6. pair solution of corresPonding to 27. Find the ordered y--3x+1 Find the ordered pair solution of x=0. 28. Find the ordered pair solution of y= lx - 5 corresponding to X=0. Find the ordered pair solution of y = f,x + tr 2 corresponding to Find the ordered pair solution of Y - -t, -, x-12. corresPonding to 300 UNIT 8 Graphs and Linear Equations 1.3 Application Problems Solve. 1. A physics student, measuring the amount of frictional force on a smooth surface, recorded the following ordered pairs: (1 ,2), (2,3), (3,5), (3,6). The first component of an ordered pair is the distance in feet the object slid across the sur- 2. perature in a town. The results were recorded as the following ordered pairs: (-3,3), (-1,2), (0,4), (1,1). The first component of an ordered pair is the temperature in degrees and the second compo- face, and the second component is the force in pounds used to move the object. Graph the For ten consecutive winter days, a meteorologist measured the tem- nent is the number of days that tem- perature was recorded during the ten-day period. Graph the ordered ordered pairs. pairs. a E C f Number of 5 days o o_ 4 C o o o e Temperature ()o I o @ o (in degrees) o 2 LL 12345 E a L ! 6 c o C o O Distance (in feet) 3. The magnification of an object at various distances from a lens was recorded as the following ordered pairs: (3,5), (s,10), (7,15), (9,20). The first component of an ordered pair is the magnification and the second component is the distance in centimeters. Graph the ordered pairs. from the object The speed of a ball as it rolls down a ramp is recorded as the following ordered pairs: (0,0), (1,2), (2,4), (3,6) The first component of an ordered pair is the time in seconds and the second component is the speed in feet per second. Graph the ordered pairs. p o 25 E 5 oo) Q) :20 := o o 4. U) 4 o) QA\ 15 a6 C g10 .2 o5 o o J 2 1 357 Magnif ication 12345 Time (in seconds) UNIT SECTION 2 2.1 Obiective 8 301 Graphs and Linear Equations Graphs of Straight Lines To graph an equation of the fotrn 1t = mx I b ol an equation in two variables is a drawing of the ordered pair solutions of the equatron. For a linear equation in two variables, the graph is a straight line. To graph a linear equation, find ordered parr solutions of the equation. Do this by The graph choosing any value of x and f inding the corresponding value of y. Repeat this procedure, choosing different values for x, unttl you have found the number of solutions desired. Since the graph of a linear equation in two variables is a straight line, and a straight Iine is determined by two points, it is necessary to find only two solutions. However, it is recommended that at least three points be used to insure accuracy. Graphy-2x+1. o () I o o o o E uo ! c E o o O Choose any values of x and then find the corresponding values of Y, The numbers 0, 2, and -1 were chosen arbitrarily for x. lt is convenient to record these solutions in a table. The horizontal axis is the x-axis. The vertical axis is the y-axis. Graph the ordered Pair solutions (0,1), (2 5), and (-1,-1) Draw a line through the ordered pair solutions. 0 2 1 2x+1 2.0 +1 2.2 +1 2(-1) +1 1 tr 1 -l 5 4 3 2 2345 1 -2 -3 r4 :5 Remember that a graph is a drawing of the ordered pair solutions of the equation' Therefore, every point on the graph is a solution of the equation and every solution ol the equation is a point on the graph. (-2,-3) and (1,3) are points on the graph and that these points are solutions of the equation Y 2x + 1 Note that Y=2x+ - 1,3) r l-1 -Z ! -4 1: I UNIT 8 302 Graph Step Step Graphs and Linear Equations y=lx-1. 1 2 Find at least three solutions, When m is a fractron, choose values of x that will simPlifY the evaluatton. Display the ordered pairs in a table. -'l 0 3 0 -3 -2 Graph the ordered pairs on a rectangular coordinate system and draw a straight line through the points. Examplel Graphy-3x-2. Example Your 2 Graph solution y - 3x + y -2 0 'l oo 1 tr -1 I 4 2 1. a o 6 E uo ! c e Example4Graphy--2x. Example3 Graphy-2x. Your Example5 Graph Solution x 0 2 I y=f,x-1. y Example6 Graph v Your solution y y=lx-3 y -1 0 -2 -2 4 solution a c o (-) 1 ct) a Gi c o o c .9 = o at UNIT 8 2.2 Objective 303 Graphs and Linear Equations To graph an equation of the forrn Ax I By - C An equation in the form Ax I By : Q, 2x a 3y = 6 (A:2,8where A, B, and C are constants, is x - 2y - -4 (A:1,8(A - 2,8 also a linear equation. Examples of 2x I y : g 4x - 5y = 2 (A-4,8these equations are shown at the right. 3, C: 6) -2, C : -4) 1, C:0) -5,C :2) To graph an equation of the form Ax + By = C, first solve the equation lor y. Then follow the same procedure used for graphing an equation of the form f - mx * b. Graph 3x + 4y : 12. 3x-y4y:12 Step 1 Solve the equation for y 4y--3x+12 a v=-:X+3 o O Step 2 Find at least three solutions. Display the ordered pairs in a table. Step 3 Graph the ordered pairs on a rectangu- lar coordinate system and draw a straight line through the points. = o @ o @ d E o I ! c 7o C oo The graph of an equation in which one of the variables is missing is either a horizontal or a vertical line. The equation y =2 could be written 0'x + | = 2. No matter what value of x is chosen, y is always 2. Some solutions to the equation are (3,2), 1-1,2), (0,2), and (-4,2). The graph is shown at the right. The graph ol y = b is a horizontal line passing through point (0,b). The equatiotl X = -2 could be written x + 0 . y = -2. No matter what value of y is chosen, x"is always -2, Some solutions of the equation are (-2,3), (-2,-2), (-2,0), and (-2,2). The graph is shown at the right. The graph ol x = a is a vertical line through point (a,0). passing 304 UNIT Example 7 Graph 2x - 8 5y Graphs and Linear Equations :10. Example Solution 2x - 5y :10 8 Graph 5x - 2y = 10. Your solulion -5y=-2x+10 y=tx-2 yv -2 0 0 5 -5 -4 Example 9 Solutlon xq2y=6. xq2y=6 2y--x+6 Example Graph Y 10 Graph x - 3y - 9. Your solution - -lx + e X 0 2 -2 4 3 2 4 o O I 1 i I : : - :-a @ o @ d E u Example 11 Solutlon Example 13 Solution Graph y = -2. The graph of an equation of the lorm y = b is a horizontal line passing through point (0,b). Graph X = 3. The graph of an equation of the form x = a is a vertical line passing through point (a,O). Example 12 Graph y :3, Your solution Example oc o E o F d 14 Graph X = -4. Your solutlon gt !i ci tr o o E B !o o UNIT 8 305 Graphs and Linear Equations 2.1 Exercises Graph. 1. Y '2x-3 3. y=I, 5. y =Z^ 4, y--3x - 1 6. y=]x+2 ::4 t-2: 7. v--f,x+z v . ,ll:4 i i i r4l y--]x+r 306 UNIT I Graphs and Linear Equations Graph. ::4 t-2 11. t y-2x-4 12. =a :-?. i ! _"..t 15. y=-!x+ ,'! i-.?. : '-..l '- -i_ _j.._ 16. f=5x-4 UNIT 8 307 Graphs and Linear Equations 2.2 Exercises -4 '-2 0 i. ...i. t. -4 19. 2x + 3y :6 20. 3x+2y-4 v v . 21. x-2y=4 :... 1.. 1.......1......! a 22. x -3y =6 308 UNIT I Graphs and Linear Equations Graph. 214 27. 29. X=3 '>4 :-2 i0 : -",i,, .-.i'-.'-. n - ;-"- 30. y=-4 .>4 .-2 -"i-.-:-i-- ! i -:-C 4x-3y=12 UNIT I SECTION 3 3.1 Obiective Graphs and Linear Equations 309 Intercepts and Slopes of Straight Lines To find the x- and y-intercepts of a straight line The graph of the equation 2x + 3y = 6 is shown at the right. The graph crosses the x-axis at the point (3,0) This point is called the x-intercept. The graph also crosses the y-axis at the point (0,2). This point is called the y-intercept. Find the x-intercept and the y-intercept of the graph of the equation 2x To find the x-intercept, let y: O. x-coordinate 2x-3y=12 : 12 O.) 2x-3y-12 2(0)-3y:12 -3y - 12 2x-3(0)=12 2x=12 x -G y (6,0). a O The x-intercept is I o @ o Find the y-intercept of A - -a The y-intercept is (0,-4). y - 3x + 4. Y:3x+4 o 6 E 3y To find the y-intercept, let x = 0, (Any point on the y-axis has (Any point on the x-axis has y-coordinate 0.) Letx=0. o I ! c - =3(0)+4 -4 c The y-intercept is (0,4). o O For any equation of the form y : mx + b, the y-intercept is (0,b). A linear equation can be graphed by finding the x- and y-intercepts and then drawing a line through the two points. Example 1 Find the x- and y-intercepts for - 2y - 4. Graph the line. Example 2 x Solution x-intercept: x-2y-4 x-2(0)=4 X=4 y-intercept: x-2y=4 0-2y-4 -2y-4 y (4,0) - -L Find the x- and y-intercepts for - y - 4. Graph the line. 4x Your solution a (0,-2) c o rt q C o o t oo UNIT 310 Example 3 Solution 8 Graphs and Linear Equations Find the x- and y-lntercepts for - 2x - 4. Graph the line. Example 4 y x-intercept: y-2x-4 0'-2x-4 -2x - -4 y-intercept: (0,b) Find the x- and y-intercepts for Graph the line. y - 3x - 6. Your solution b- -4 (0, X=2 -4) (2,0) 4 o € ti ci c o -g 6 at 3.2 Obiective To find the slope of a straight line Thegraphs oIy-!x+l andy=2xq.1 are shown at the right. Each graph crosses the y-axis at the point (0,1), but the graphs have different slants. The slope of a line is a measure of the slant of a line. The symbol for slope is m. The slope oJ a line containing two points is the ratio of the change in the y values of the two points to the change in the x values. The line containing the points (-2,-3) and (6,1) is graphed at the right. The change in the y values is the difference between the two ordinates. Changeiny- 1 -(-3)=4 The change in the x values is the difference between the two abscissas. Changeinx:6 -(-2)=B changeiny 4 changeinx B Slope=ffi=-.L 1 2 A formula for the slope of a ltne containing two points, P, and Pr, whose coordinates are (x,,y-') and (xr,yr), is given by: Slooe=m=fz-ft ' x2-xt UN lT I Graphs and Linear Equations 311 Find the slope of the line containing the points (-1,1) Let P., = (- 1 ,1) and P2 = (2,3). (lt does not matter which point is named P., or Pr', the slope will be the same.) v^-v. Xz-X, 2 -(-1) and q,3) ;.i ,,, j . 3 positive slope A line which slants upward to the right always has a positive stope. Find the slope of the line containing the points Let P, - (-3,4) and (-3,4) and (2,-2). P2: (2,-Z). -2-4 ^-fz-ltXz-Xt - 2-(_3) -6 =_=__ 5 y 6 S oo :I negative slope o @ o @ d A line which slants downward to the right always has a negative slope. E o oc 6 C o c o o Find the slope of the line containing the points Let P. = (-1,3) and P2 (-1,3) and (4,3). = @,3). 3;3, Jn-fz-Yt -9-O Xz-Xr = 4-\-1) b zero slope A horizontal line has zero slope. Find the slope of a line containing the points = (2,-2) and Pr: (2,4). 4. (-.2) , -Yz-lr Xz-xt- 2-2 -o Nota (2,-2) and (2,4). Let P., 0 A vertical line has no slope. l- l-(2,4)^i i real i -i --i i:.." i tii numOei no slope UNIT I 312 Example 5 Graphs and Linear Equations Example Find the slope of the line containing the points (-2,-1) and 6 = (-2,-1) Let P.' P2: (-1,2) and (1,3). (3,4), Solution Find the slope of the line con- taining the points Your solution and (3,4). 4-(-1) 5 ^ '-x._-\-3-GA-5-' -Yz-lt - 1 The slope is 1. Example 7 Find the slope of the line containing the points (-3,1) and Example 8 (2,-2). Solution Let P., = Find the slope of the line containing the points (1,2) and (4, (-3,1) Ihe slope,t Your solution and P, = (2,-2). v--v.t1 t2 n -2-1 - Xz-Xt - 2-(-3) -5) -3 5 oo -;. I @ o o Example 9 Find the slope of the line containing the points (-1,4) and (- Solution 1 Example 10 Find the slope of the line containing the points (2,3) and (2,7). ,0). Let P, = (-1,4) and P, = (-1,0). 0-4 ^ -Yz-Yt "'-lf,i--1-i:t- c t o I E C G c s) oo Your solution -4 o The line has no slope. Example 11 Find the slope of the line containing the points (-'1,2) and Example 12 (-5,-3). (4,2). Solution Let P, P' = - (-'1,2) and Find the slope of the line con- taining the points (1,-3) and Your solution (4,2). ^=';+=ffi=$=o The line has zero slope. o aI ct (a !t a c o 6 c : .9 E at UNIT 3.3 Obiective Graphs and Linear Equations 8 313 To graph a line using the slope and y-intercept - 3, * t ,. shown at the right. The points (-3,-1) The graph of the equation y and (3,3) are on the graph. The slope of the line is: 3-(-1) ,',__ - _ 3-(-3) _46 _23 Note that the slope of the line has the same value as the coef{icient of x. | : For any equation of the form trlx 1 b, the slope of the line is m, the coefficient of The y-intercept is (0,b). Thus, the torm y = fftX b is called the slope-intercept x. * form of a straight line. Find the slope and the y-intercept of the line v 'o o @ o d E Slope I! 6 c Ic -]x + t. =filx + [t * '= l-* l" L! I o y= = n1 The slope is 3 -4' l- 1 = -q y-intercepl = (0, b) = (0, 1) The y-intercept is (0,1) o O When the equation of a straight line is in the form y = mx * b, the graph can be drawn using the slope and y-intercept. First locate the y-intercept. Use the slope to find a second point on the line. Then draw a line through the two points. Graphy-2x-3. y-intercept : (0,b) = (0,-3) m_D_?_changeiny I"-L -1-;r*q., Beginning right at the y-intercept, 1 unlt (change in move x) and then up 2 units (change in y). (1 , - 1) is a second pornt on the graph. Draw a line through the two points -3) and (1 , - 1 ). (0, (1 , -1) lup2 J;i i 314 UN Example 13 Graph lT 8 Graphs and Line4r Equations y- -!x+l byusing Example 14 the slope and y-intercePt. the slope and y-intercePt. Solution Your solution (0,b) = (0,1) (Move right 3 units, -3- then down 2 units.) y-intercept 'It Graphy=-Ir-lbyusing = 2 - -3 - -2 v risht3l I : i-ldown Example 15 Graph y - -1r I : OU using the Example 16 y-intercept : ,tr--4- 3_-3 y - -3" OU using the slope and y-intercePt. slope and y-intercePt, Solution Graph (0,b) = (0,0) Your solution _a 4 t o @ -4 6 F I 1) c .....j.;.... i...... i c o c o o 0 ,:-3 -2: -1 -.i ::i; i' i i -i-2 < Example 17 Graph 2x - 3y - 6 by using Example 18 Solutlon Solve the equation for y. Graph x - 2y - 4 by using the slope and y-intercePt. the slope and y-intercept. Your solution 2x-3y=6 -3y=-2x+6 v=lx-2 y-intercept v = (0,-2) ^ =3 o l' ri c o o t o o o UN lT I Graphs and Linear Equations 315 3.1 Exercises Find the x- and y-iniercepts 1. X-L=3 2. 4' y=2x+10 5. x-5y=10 6. 7. y-3x+12 8. Y:5x+10 9. 2x-3y-O 10. 3x+4y-0 11. 3x + 4y :12 y--!x+s 3' l:3x-G 12. o O I o o 6 Find the x- and y-intercepts and graph. E o L ! z 13. 5xa2y=10 @ v c o O 15. y=|x-3 ::q i2 :-4 i*i 14. x-3y=6 3x + 2y :12 y:lx-a UNIT 8 316 Graphs and Linear Equations 3.2 Exercises Find the slope of the line containing the points. 17. P{4,2), P2G,4) 18. P1Q,1), P2G,4) 19. P,(-t,3), 20. P{-2,1), P2Q,2) 21. P{2,4), P2(4,-1) 22. 23. P{-2,3), P2(2,1) 24. P{5,-2), P,(1,0) P1(1,3), P2(2,4) P2(5,-3) 25. Pl(8,-3), P29,1) o O z -o @ o 26. P1(0,3), PzQ,-1) 27. P{3,-4), P13,5) 28, P,(-r o ,2), P2G1,3) ( E Io ! 6 c c o o 32. P{4,-2), P2(3,-2) 30. P1(3,0); P2(2,-1) 33. P{-2,3), P2(J,3) P1(5,1), P2G2,1) 35. P{-2,4), P2G1,-1) 36. P1(6, 38. P.,(-1,5), P,(5,1) Pl(5,1), P2(5,-2) 39. -4), P2G,-2) 31. P,(0, - 1), P2(3,-2) 34. Pr(4,-1), Pr(-3,-1) 37. P,(-2,-3), 40. P,(- P2(-2,1) t ,5), P2Q ,1) UNIT 8 317 Graphs and Linear Equations 3.3 Exercises Graph by using the slope and y-intercept. 41. y-3x+1 44. y 45. 2xay=3 47. x-2y=4 =|x + 1 46. 3x-y=1 318 s0. , =L^ o o I o o o E uo E C o c oo 56. UNIT 8 SECTION 4 319 Graphs and Linear Equations Equations of Straight Lines 4.1 Objective To i"d the equation of a line using the equation):mx * b When the slope of a line and a point on the line are known, the equation of the line can be written using the slope-intercept form, y - mx a b. Find the equation of a line which has slope 3 and y-intercept (0,2). The given slope, 3, is Replace m with 3. m. y-mxqb y-3x+b l The given point, (0,2), is the y-intercept. Replace b with rr-?vtD v,\ ,/ 2. Find the equation of the line which has slope I - | L and contains point (-2,4) y-mx1b The given stope, o o I ], y:Lr+o is m. Reolace m with 1. '2 @ o c The given poinl, (-2,4) is a solution of the equation of the line. Replace x and y in the equation with the coordinates of the point. o O Solve for b, the y-intercept. o 6 E o L ! a 6 Ic 4=t(-2)+o 4=-l 5=b y-mxab Write the equation of the line by replacing m and b in the equation by their values. Example 1 (3,-3) Solution t Find the equation of the line which contains the point and has sloOe t l=ix+o -s:fta)+o -3=2+b tr_A f +O Example 2 - Find the equation of the line which contains the point (4,-2) , 1_ vf | va, 2'\ and has stope |. Your solution (\l ao tc t - t_ J^'\ tt c o g o =o tt UNIT 920 4.2 Obiective 8 Graphs and Linear Equations To find the equation of a line using the point-slope formula An alternate method for finding the equation of a line, given the slope and a point on the line, involves use of the point-slope formula. The point-slope formula is derived from the formula for slope. Let (x.',/r) be the given point on the line and(x,l) be any other point on the line. r t1 _ h Itt X-Xt - Formula for slope Yr- Multiply both sides of the equation by (x - x.,). Y,r. X-Xr- (r -- xr) - ntrx - x,) y-yt-m(x-xr) Simplify. = m(x 'x,tl. ln this equation, m is the slope and (xt,y) is the given point. The point-slope lormula is y - lt Find the equation of a line which passes through point (x.,,y.) = (-3,2) (-3,2) and has slope l-h-m(x-xr) y -z =f t" - <-slt ^ =3 oo I o @ y-2:!1x+3) I o y-2=lx+2 y The equation of the line is Example 3 y= to line point (-2,-l) and has slope f . Use the point-slope formula find the equation of a which passes through (x.,,y.,):(-2,-1) y-yt-m(x-xr) Solution, =* =!x + f,. G E I ! c a c o c o o lx + A Example 4 Use the point-slope formula to find the equation of a line which passes through point g,-2) and has stooe f . Yoursolution y_(_1)=8tx_(_2)l r*1=|fr+21 Y+1-|x+s y=|x+2 The equation of the line \/ 'z :1x + 6t o $ is ri c o z. o 3o at UNIT Graphs and Linear Equations 8 321 4.1 Exercises Use the slope-iniercept form. 1. Find the equation of the line which contains the point (0,2) and has slope 2. 3. Find the equation of the line which contains ihe point (-1 ,2) and has 2. slope 4. slope *3. o O 5. Find the equation of the line which contains the point (3,1) and has I o o stope o Find the equation of the line which contains the point (0, 1) and has - - 2. Find the equation of the line which contains the point (2,-3) and has slope 3. 6. I Find the equation of the line which contains the point (-2,3) and has slope ]. E o L ! C c o c o O 7. Find the equation of the line which contains the point (4,-2) and has stope 9. 11. - Find the equation of the line which contains the point (2,3) and has stope f; Find the equation of the line which contains the point (5, -3) and has stope 8. 10. Find the equation of the line which contains the point (5, i ) and has - stope f;. Find the equation of the line which - |. 12, |, contains the point (2,3) and has Find the equation of the line which contains the point (-1,2) and has slope slope ]. -| 322 UN lT 8 Graphs and Linear Equations 4.2 Exercises Use the point-slope formula. 13. 15. 17. Find the equation of the line which passes through point (1, -1) and has slope 2. 14. Find the equation of the line which passes through point (-2,1) and has slope -2. 16. Find the equation of the line which 18. Find the equation of the line which passes through the point (2,3) and has slope -.l. Find the equation of the line which passes through point and has slope (-1,-3) -3. Find the equation of the line which o passes through point (0,0) and O passes through point (0,0) and >t TI has slope has slope o\ o f,. - |. @ d E uo E C 6 o c o O 19. Find the equation of the line which passes through point (2,3) and has slope 21. 23. -f a f . 22, 24. f,. Find the equation of the line which passes through point (-5,0) and has slooe . Find the equation of the line which passes through point (-2,1) and has slope Find the equation of the line whtch passes through point (3,-1) and has slope |. Find the equation of the line which passes through point (-4,1) and has slooe 20. - ]. Find the equation of the line which passes through point (3,-2) and has slope |. UNIT 8 Review SECTION 1 1.1 323 Graphs and Linear Equations /Test Graph the ordered pairs and (0,2). (-3,1) Find the ordered pair solulion of 1.2 y u^ - -|xJ + 2corresponding v_o 1.3 The costs for an amount of energy were recorded as the follow- ing ordered pairs: (1 ,5), (2,10), (4,20), and (5,25), where the abscissa is the number of kiloI watt-hours used and the ordinate is the cost in cents. Graph the o ordered pairs. o o a 25 o 20 c C) c u) o O 15 10 E 0 12345 Kilowatt-hours o E uo oC c Ic o c) SECTION 2 2.1a Graphy-3x+1. 2.1b Graphy--]x+s. 2.2b Graphx*3=0. to 324 UNIT I Graphs and Linear Equations Review SECTION 3 /Test 3.1a Find the x- and y-intercept for 6x-4y-12. Y=*x+t' Find the slope of the line contain- ing the points (2,-3) 3.1b Find the x- and y-intercept for 3.2b and (4,1). Find the slope of the line containing the points (3, -4) and (1 , 3.3a Graph the line which has slope - anO y-intercept (0,4). f SECTION 4 4.1a Find the equation of the tine which contains the point (0,-1) and has slope 3. 4.2a Find the equation of the line which contains the point (2,3) and has stope |. 3.3b -4). Graph the line which has stope 2 and y-intercept -2. 4.1b Find the equation of the line which contains the point (-3,1) and has slope $. 4.2b Find the equation of the line which contains the point (-1,2) and has stope -f . UN lT 8 325 Graphs and Linear Equations Review/Test SECTION 1 1.1 Which point (-2,-3)? coordinates has 1.2 a)A b)B a) b) c) d) d)D 1.3 o o 2 (-2,3) (-2,-5) (-2,5) (-2,-3) : A physicist recorded the distances a ball would rolldown a ramp as the following ordered pairs: (1,1), (2,5), (3,7). The abscissa istime in seconds and the ordinate is distance in feet. What is the distance the ball rolled in 2 s? b) 3ft a) 1ft SECTION o A--L. c)c :3-2-1ol 1 2 3; , 1l : ' : ,I:: eQ,,-21 Find the ordered pair solution of - 2x - 1 corresponding to y 21a Which is the y-2x-1? a)yb) graph c) sft of d) 2.1b Which is the graPh ,r-. Y -^1 a)yb)Y v 7t1 t-lJ I o @ d E o I o c 6 c o o O 2.2a a) Which is the 2x-3y -6? v b)v graPh ot 2.2b a) Which ts y-3=0? the b) graph v of UNIT 8 326 Graphs and Linear Equations Review/Test SECTION 3 3.1a 3.2a Name the x-intercept for the line 2x-3y=12. a) (6,0) c) (0, -3) b) d) (0, -4) 3) a)l c)5 b) 7 5 d) -7 Name the y-intercept for the line 5x+2y=20. a) c) (2,0) Find the slope of the line containpoints (2, ing and ( 3,4). - the 3.1b 3.2b (0,2) (5,0) b) d) (0,10) (4,0) Find the slope of the line contain- ing the points (2,3) and a)4 c) no slope (-2,3). b) -4 d)0 5 Which is the graph of the line which has slope -1 and y-intercept (0,2)? vb) 3.3b Which is the graph of the line which has stoOe and y-intercept v (0,-l)? I vb) a) 0 SECTION 4 4.1a Find the equation of the which contains the point and has slope -3. a) y = -3x a2 b) y-bx-2 c) Y:2x-3 d) l:*2x+3 line (Q,2) 4.1b Find the equation of the line which contains the point (2,-1) and has stope |. a) y_rx_1 - .t D) 13 y=Zx*z c),15Y=Zx-fz .. y_rx_2 d) 1 4.2a Find the equation of the line which contains the point (6,1) and has slope ? 3' .t a) y:ix+1 b) y:tr+a c) y=lx-3 d) v=?r-t 4.2b Find the equation of the line which contains the point and has slope 2. a) y=2x-1 b) y=2xt2 c) y =2x d) y=2x-3 (-1,0)