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Itg2 U2 Act.1 10/2/2011 Unit 2 – Parallel and Perpendicular Lines Activity What do you notice about these lines? Part 1 1. The following equations are graphed to the right Line A: y = 3x + 2 Line B: y = 3x + 4 Identify each line and label it correctly (A or B) a) What do you notice about the lines? b) What is the slope of Line A? c) What is the slope of Line B? 2. Can you make any connection about the lines and their slopes? 3. Write the equations of two lines that are parallel. 4. The following equations are graphed to the right Line A: y = -3x + 5 Line B: a) What do you notice about the lines? b) What is the slope of Line A? c) What is the slope of Line B? 5. What conjecture can you make about slopes and perpendicular lines? 6. Find the negative reciprocal of the following numbers: a) 5 b) c) -3 d) e) 1 7. Write the equation of two sets of lines that are perpendicular to each other. Itg2 U2 WS2.1 Unit 2 – Parallel and Perpendicular Lines 1. a) Define parallel. b) Define perpendicular. 2 What do you know about the slopes of parallel lines? 3. What do you know about the slopes of perpendicular lines? Itg2 U2 WS2.3 Integrated Math 2 – Parallel Lines WS #2-3 1. How can you tell if two lines are parallel before graphing them? For each problem below, write the equation of the line in all three forms and graph BOTH lines to make sure you are correct. For each set of equations below, tell whether the two lines are parallel to each other, perpendicular to each other or neither. 2. A line that passes through the point (-2, 5) and is parallel to the line y = x + 2 4. y = 2x – 5 and y = 2x + 8 5. y = 1/4 x + 3 and y = - 4x – 3 3. A line that passes through the point (1, 4) and is parallel to the line y = -3x + 1 6. y = - 2x + 8 and y = - 1/2 x - 10 7. y = 12x – 1 and y = 12x + 3 8. y = 1/2 (x – 8) + 3 and y = - 1/2 (x + 4) – 1 9. 2 y = /3 (x + 6) – 2 3 and y = /2 (x – 2) + 9 10. y = 2/3 x – 7 and y = - 3/2 x + 9 11. y = - 4x and y = 1/4 x – 4 4. A line that passes through the point (0, 2) and is parallel to the line y = 5x – 6 5. A line that passes through the point (-6, 1) and is parallel to the line y = 1/2 x – 2 6. A line that passes through the point (-27, -12) and is parallel to the line y = 2/3 x – 2 Itg2 U2 WS2.2 7. A line that passes through the point (10, 3) and is Unit 2 – Slope-Intercept form (Solve for Y) parallel to the line y = 3/5 x – 1 1. Which of the three forms of equations do you like best? Why? Itg2 U2 WS2.4 2 To find the slope which form would you prefer? Integrated Math 2 – Perpendicular Lines WS #2-4 Rewrite each equation below in slope-intercept form (in other 1. a) What does opposite reciprocal mean? words, solve for y). Then write the equation of ANY line that is b) What types of slopes does it refer to? parallel to the given line. And then write the equation of ANY line which is perpendicular. You may write your equations in slopeintercept form. 3. 4x + y = 11 4. x–y=3 5. 5x – 10y = 9 6. -3x + 2y = 8 7. -6x – 3y = 9 8. 9x + 2y = 16 9. -7x – 2y = 10 10. x + 5y = 15 For each problem below, write the equation of the line in all three forms (You may wish to Graph BOTH lines to make sure you are doing them are correct). 2. A line that passes through the point (-1, 4) and is perpendicular to the line y = x + 2 3. A line that passes through the point (6, 2) and is perpendicular to the line y = -3x + 1 4. A line that passes through the point (0, 2) and is perpendicular to y = 5x – 6 5. A line that is perpendicular to y = 1/2 (x – 2) – 4, and passes through the point (-1, -1) 6. A line that is perpendicular to -2x + 3y = 9, and passes through the point (-12, 7) 7. A line that is perpendicular to 4x + y = – 1, and passes through the point (-8, -3) Itg2 U2 PQz1 - Prep1 for Quiz Fall 2011 Itg2 U2 PQz1.2 - Prep2 for Quiz Fall 2011 Integrated Math 2 – Unit 2 Practice Quiz – || and Integrated Math 2 – Unit 2 Practice Quiz 2 – || and For questions 1-4 below, determine if the lines are parallel ( || ), perpendicular ( ), or neither and explain why. (3 points each) For questions 1-4 below, determine if the lines are parallel ( || ), perpendicular ( ), or neither and explain why. (3 points each) 1. y = 3x – 5 and y = -5(x + 1) – 2 1. y = 2/3 (x – 6) + 3 and y = 2/3 x – 4 2. -3x + 5y = 15 and y = - 5/3 x + 6 2. y = 4/3 x – 2 - 3x + 4y = 20 3. y = 2/3 (x – 6) + 4 and y = 2/3 x – 5 3. y = - 4(x – 2) – 5 and y = 3x – 7 4. -4x + y = 7 y = 1/4 x – 4 4. y = - 3/2 x + 6 - 2x + 3y = - 15 5. a) What does opposite reciprocal mean? (2 points) 5. a) What does opposite reciprocal mean? (2 points) and and and b) What types of slopes does it refer to? b) What types of slopes does it refer to? Rewrite each equation below in slope-intercept form. Then write the equation of any line that is parallel to the given line AND perpendicular to the given line. (4 points each) Rewrite each equation below in slope-intercept form. Then write the equation of any line that is parallel to the given line AND perpendicular to the given line. (4 points each) 6. 3x + 4y = 20 6. 2x + 5y = - 10 7. 7x – 8y = -80 7. 7x – 4y = 12 For each problem below, write the equation of the line in all three forms. GRAPH BOTH LINES to check your answers. (7 points each) For each problem below, write the equation of the line in all three forms. GRAPH BOTH LINES to check your answers. (7 points each) 8. A line that passes through the point (2, 8) and is parallel to the line y = 3x – 6 8. A line that passes through the point (8, -2) and is parallel to the line y = - 3/4 x – 5 9. A line that passes through the point (-9, -5) and is perpendicular to the line y = -3x – 7 9. A line that passes through the point (6, 4) and is perpendicular to the line y = -3x + 1 10. A line that is parallel to the line y = - 1/4 x + 1, and passes through the point (2, 2) 10. A line that is parallel to the line y = 2x + 4, and passes through the point (3, 5) 11. A line that is perpendicular to the line y = 1/3 x – 2, and passes through the point (-4, 8) 11. A line that is perpendicular to the line y = 1/2 x – 5, and passes through the point (2, -3) Itg2 U2 Qz1 - Quiz 1 Fall 2011 Integrated Math 2 – Unit 2 Quiz – Parallel/Perpendicular For questions 1-4 below, determine if the lines are parallel ( || ), perpendicular ( ), or neither and explain why. (3 points each) 1. y = 2x – 1 and y = - 3(x – 1) – 4 2. - 2x + 3y = -12 and y = - 3/2 x + 7 3 y = /4 (x – 8) + 1 and y = /4 x – 1 4. - 2x + y = 3 y= 5. For each system of equations below, find the solution by graphing. Write your answer as an ordered pair. 2. y= x and y= -x+2 3. y = 2x – 6 and y = -2x – 2 4. y = -x + 1 and y = 3x + 5 5. y = 3(x – 1) + 4 and y = 1/2 x + 1 6. y = 2(x + 3) – 5 and y = - 3x – 9 7. y = 2x – 1 and y=-x+ 8 8. y = (x – 4) + 2 and y = - 3x + 2 9. y = 1/2 x – 3 and y = 3/2 x – 1 3 3. and Itg2 U2 WS2.5 Day 1 Graphing 1. What does the solution look like on a graph for a system of equations? 1 /2 x – 3 a) What does opposite reciprocal mean? (2 points) b) What types of slopes does it refer to? Rewrite each equation below in slope-intercept form. Then write the equation of any line that is parallel to the given line AND perpendicular to the given line. (4 points each) Itg2 U2 WS2.6 Day 2 Graphing Integrated Math 2 – Graphing Systems WS #2-6 1. What does the solution look like on a graph for a system of equations with no solutions? 6. 2x + 3y = 15 7. 5x – 3y = -9 For each problem below, write the equation of the line in all three forms. GRAPH BOTH LINES to check your answers. (7 points each) 8. A line that passes through the point (3, 7) and is parallel to y = 2x – 3 9. A line that passes through the point (-2, -4) and is perpendicular to the point y = -2x – 6 2. What does the solution look like on a graph for a system of equations with infinite solutions? For each system of equations below, find the solution by graphing. Write your answer as an ordered pair. 3. y = 1/2 x – 3 and x – 2y = 6 4. y = 2x – 6 and -2x + y = 3 5. y=-x+1 and y = 1/2 x + 4 6. y = 2(x – 4) + 3 and -2x + y = 1 7. y = 3x + 3 and 6x – 2y = - 6 8. y=-x+6 and y = 1/2 x + 3 9. y = (x – 1) – 5 and y= x–2 10. 4x + 3y = 12 and 2x – 3y = 6 1 10. A line that is parallel to the line y = - /3 x + 1, and passes through the point (9, 3) 11. A line that is perpendicular to the line y = 1/2 x + 3, and passes through the point (4, 0) Itg2 U2 WS2.7 Day 1 Substitution 1. What will indicate to you that you should be using substitution to solve a system of equations? Solve each system of equations below by using substitution. 2. y = 2x – 6 and 3x + 2y = 9 3. 2x + 4y = 8 and y = 2x + 13 4. 3y – 2x = 11 and y = 9 – 2x 5. x = 3y – 2 and -x + 4y = 4 6. x+y=-9 and x = 2y 7. y = 2x – 8 and x = - 3y – 3 8. x = - 3y + 16 and y = - 2x + 7 9. y = - 2x – 14 and y = x + 10 Itg2 U2 WS2.9 Day 1 Elimination Solve by elimination 1. Describe a problem that would be easiest to solve by elimination. (What form are these problems in?) 2. 3y – 2x = 11 and y + 2x = 9 3. 2x + 3y = -1 and 5x – 3y = 29 4. 4x + 5y = 33 and - 4x – 3y = -23 5. 5x – 3y = 6 and 2x + 3y = 15 and 3x + y = 9 6. 4x – 3y = - 1 7. 2x + 5y = 4 and 3x – 10y = -29 8. and 4x – 2y = -26 and 3x – 4y = 5 3x + 4y = 8 9. 2x + 5y = 11 Itg2 U2 WS2.10 Day 2 Elimination Itg2 U2 WS2.8 Day 2 Substitution 1. What does it mean if you get a solution of 3 = 3 when doing substitution? Explain. 2. What does it mean if you get a solution of -2 = 5 when doing substitution? Explain. Solve by elimination 1. When you get a solution of 4=4 when solving by elimination, what does the graph of the equations look like? Explain. 2. When you get a solution of -2 = 2 when solving by elimination, what does the graph of the equations look like? Explain. Solve each system of equations below by using substitution. 3. y = x + 13 and 2x – y = - 20 4. x = 3y – 2 and 6y = 2x + 36 5. y = - 2/3 x + 1 and 2x + 3y = 3 6. x = -2y + 4 and 6y + 3x = 12 7. y= x–9 and x = 2y + 4 8. 3x = 5 – y and y = - 3x + 1 9. x = - 5y – 1 and x = 3y – 33 10. y= -x+5 and y = 2x – 34 Solve by using substitution. 3. x + y = 4 and 2x + 2y = 4 4. 2x + 14y = 10 and x + 7y = 5 5. 6x – 5y = 31 and 3x + 2y = 20 6. 2x – 3y = - 2 and 4x + y = 24 7. 2x + y = 8 8. 2x + 5y = 11 9. 3x + 2y = 12 10. 6x – 3y = 12 and and and and 4x + 2y = 12 3x – 4y = 5 5x – 3y = 1 2x – y = 4 Itg2 U2 WS2.11 Use the data below to create a scatterplot. Then draw a line of best fit. Make sure to label your graph. Integrated Math 2 – all 3 methods 1. What does the solution of a system of equation look like on a graph? As an answer? Solve each system of equations below by graphing. 2. y = 1/2 x – 3 and y = 1/2 (x – 8) + 1 3. y = 3x - 3 and y = 3x + 2 Solve each system of equations below by using substitution. 4. y = 4x – 2 and 6x – 2y = - 2 5. 5x – y = 8 and x = 10 – 4y Solve by using elimination. 6. 2x – 8y = -18 and x + 8y = 15 7. 2x + 3y = 2 and - 5x + 2y = 33 Itg2 U2 Unit 2 Review for Test 1. What two things do you know about parallel lines? Create your own pair of parallel lines and graph them. 2. What two things do you know about perpendicular lines? Create your own pair of parallel lines and graph them. Put the following in Slope-Intercept form (Solve for y) 3. - 3x + y = - 2 4. 4x – 6y = 12 5. 2x + 2y = 4 Solve each system of equations below by graphing. 6. y = 3x – 6 and y = 1/2 x + 4 7. y = 1/2 x + 5 and y = - 2x – 5 8. y = 2x – 4 and y = 2x + 3 Solve each of the systems of equations by substitution. 9. x = 5y – 2 and 2x – 2y = - 12 10. 3x – y = 18 and y = 2x – 13 Solve by using elimination. 11. - 4x + 2y = 22 and 4x + 3y = 13 12. 3x – 5y = 23 2x + 3y = - 10 and 13. Pick two points from your line (not your data) and write the equation of the line in all three forms. 14. Use the equation of the line that you just created to answer the following questions. a) At what distance would you predict if there were 32 goals made? b) How many field goals would you predict were made from a distance of 38 yards?? Itg2 U2 WS2.1 ½ (Class-Work) Parallel and Perpendicular Lines Indicate whether the following are || , Neither || 1. y = 5x – 2 and y = 5x + 1 2. y = 1/2 x – 4 and y = 1/2 x + 3 || 3. y = - 5x – 6 and y = 1/5 x + 5 4. y = - 1/2 x – 8 and y = - 1/2 x + 7 || 5. y = 2x – 2 and y = 5x + 1 6. y = 5x – 2 and y = - 1/5 x + 1 7. y = 2/5 x – 4 and y = 2/5 x + 3 || 8. y = - 5/6 x – 6 and y = 6/5 x + 5 9. y = - 1/2 x – 8 and y = 1/2 x + 7 Neither 10. y = 2x – 2 and y = -2x + 1 Neither 11. y = 5(x – 2) + 1 and y = 5(x + 3) + 6 12. y = 2/5(x + 2) + 7 and y = - 5/2 (x – 3) + 2 13. y = - 3/4(x – 2) + 3 and y = - 3/4 (x + 3) – 3 || 14. y = 2(x – 2) + 2 and y = - 1/2 (x – 2) + 4 15. y = 3(x + 2) + 14 and y = 3(x + 2) – 3 || 16. y = 5(x – 2) + 7 || 17. y = 2/5 (x + 2) – 5 and y = 2/5 (x – 2) + 8 18. y = -6/5 (x – 2) + 4 and y = - 5/6 (x – 2) + 9 Neither 19. y = - 1/2 (x – 2) – 3 and y = - 1/2 (x + 2) + 1 || 20. y = 2(x – 2) + 2 and y = - 1/2 (x – 2) – 2 and y = 5(x – 2) + 5 Changing equations into S-I form (Solve for y). Next write two equations: (a) || and (b) Ex1. -2y = 5x – 2 2y = - 5x + 2 y = - 5/2 x + 1 || y = - 5/2 x + 1 y = 2/5 x + 3 Neither || Ex2. 4x + 2y = 8 2y = - 4x + 8 y = -2x + 4 || y = -2x + 4 y = 1/2 x – 5 Ex3. - 9x + 3y = 12 3y = 9x + 12 y = 3x + 4 || Ex4. y = 3x – 2 y = - 1/3 x + 7 7x – 3y = 12 -3y = - 7x + 12 -3 -3 7 y = /3 x – 4 || y = 7/3 x + 9 y = - 3/7 x + 2 -------------------------------------------------1) x + 6y = 12 -x -x 6y = -x + 12 6 6 1 y = - /6 x + 2 2) -2x + 3y = 9 +2x +2x 3y = 2x + 9 3 3 2 y = /3 x + 3 3) 4x – 8y = 16 -4x -4x -8y = -4x + 16 -8 -8 y = 1/2 x - 2 4) 2x + 3y = 15 3y = -2x + 15 y = - 2/3 x + 5 5) -x + 4y = 12 4y = x + 12 y = 1/4 x + 3 6) -6x – 2y = 10 -2y = 6x + 10 -2 -2 y = -3x - 5 || Itg2 U2 WS2.3 1/2 Integrated Math 2 – Parallel Lines WS #2-3 Find the equation of the line that passes through the given point and is parallel to the given line. Write the equation of the line in all three forms 1. (3, 7) y = 2x – 3 A parallel slope means that the slope is the same. m = 2 , (3, 7) y = 2(x – 3) + 7 y = 2x – 6 + 7 y = 2x + 1 -2x + y = 1 2x – y = -1 2. (-2, 5) y = x + 2 m = 1 , (-2, 5) y = 1(x + 2) + 5 y=x+2+5 y= x+ 7 -x -x -x + y = 7 x–y=-7 3. (1, 4) y = -3x + 1 m = - 3 , (1, 4) y = -3(x – 1) + 4 y = -3x + 3 + 4 y = - 3x + 7 +3x +3x 3x + y = 7 4. (0, 2) y = 5x – 6 m = 5 , (0, 2) y = 5(x – 0) + 2 y = 5x + 2 y = 5x + 2 -5x -5x -5x + y = 2 5x – y = -2 5. (-6, 1) y = 1/2 x – 2 m = 1/2 , (-6, 1) y = 1/2 (x + 6) + 1 y = 1/2 x + 3 + 1 y = 1/2 x + 4 2y = x + 8 -x -x -x + 2y = 8 x – 2y = -8 6. (9, 3) y = - 1/3 x + 1 m = - 1/3 , (9, 3) y = - 1/3(x – 9) + 3 y = -3x + 3 + 3 y = 1/3 x + 6 3y = x + 18 -x -x -x + 3y = 18 x – 3y = -18 7. (-27, -12) y = 2/3 x m = 2/3 , (-27, -12) y = 2/3 (x + 27) – 12 y = 2/3 x + 18 – 12 y = 2/3 x + 6 3y = 2x + 18 -2x -2x -2x + 3y = 18 2x – 3y = -18 8. (10, 3) y = 3/5 x – 1 m = 3/5 , (9, 3) y = 3/5 (x – 10) + 3 y = 3/5 x – 6 + 3 y = 3/5 x – 3 5y = 3x – 15 -3x -3x -3x + 5y = -15 3x – 5y = 15 Itg2 U2 WS2. B EXTRA PRACTISE (Needs to be fixed up) 1) A line contains the points (3, -1) and (-1, 2). Another line graphed in the same coordinate plane contains the points (2, 0) and (-2, 3). a) How do these two lines relate to each other? b) Write the equation of each line in any of the three forms. m1 = (- 1) – (2) m2 = (0) – (3) (3) – (-1) (2) – (-2) = - 3/ 4 4 5/5 = 1 1/5 + b – 1 1/5 – 1 1/5 3 4/5 = b y = - 3/5 x + 3 4/5 2. A line contains the points (7, 3) and (-1, -4). Find the equation of the line. = ( 3) – (- 4) ( 7) – (- 1) = 3+ 4 7+ 1 = 7/8 = - 3/ 4 y1 = - 3/4(x – 3) – 1 y = - 3/4 x + 9/4 – 4/4 y = - 3/4 x + 5/4 y = - 3/4 x + 5/4 y2 = - 3/4(x – 2) + 0 y = - 3/4 x + 6/4 y = - 3/4 x + 3/2 y = mx + b y = (7/8) x + b (3) = (7/8)(7) + b 3 = 49/8 + b Therefore the two lines are parallel, since they have the same slope and different y-intercepts 2) A line contains the points (0, -2) and (1, 3). Another line graphed in the same coordinate plane contains the points (3, -1) and (-2, 0). How do these two lines relate to each other? m1 = ( - 2) – (3) m2 = ( - 1) – (0) (0) – ( 1) (3) – (- 2) = - 5/ -1 = 3 = 7 1/8 + b – 7 1/8 – 7 1/8 - 4 1/8 = b y = 7/8 x – 4 1/8 3. A line contains the points (4, -3) and (8, -1). Find the equation of the line. = (- 3) – (- 1) ( 4) – ( 8) - 1/ 5 = -3+ 1 -4 = -2 -4 = 1/2 = 5 Therefore the two lines are perpendicular, since the slopes are the opposite reciprocal of each other. y = mx + b y = (1/2) x + b (- 3) = (1/2)(8) + b 3) A line contains the points (0, -1) and (-1, 2). Another line graphed in the same coordinate plane contains the points (2, 0) and (-2, 3). How do these two lines relate to each other? m1 = ( - 1) – (2) m2 = ( 0) – (3) (0) – (- 1) (2) – (- 2) = - 3/ 1 = 1. A line contains the points (-2, 5) and (3, 2). Find the equation of the line. m =(5)–(2) (- 2) – (3) y = mx + b y = (- 3/5) x + b (5) = (- 3/5)(- 2) + b 5 = 6/5 + b -7 = b y = 1/2 x – 7 - 3/ 4 = -3 Therefore the two lines will intersect, but they are neither perpendicular nor parallel. = - 3/5 -3 =4+ b -4 -4 2) Write the equation of a line in slope-intercept form: (a) parallel and (b) perpendicular to the line: 4x - 2y = 8 and contains the point (-6, 2). Solve both: (a) mathematically and (b) graphically. (a) Mathematical Method - 4x + 2y = - 8 (Make y positive by multiplying by -1) + 4x + 4x (Eventually get in the form: y = mx + b) 2y = 4x – 8 2 2 y = 2x – 4 Parallel Slope Perpendicular Slope y = 2x + b y = - ½ x + b [Substitute in (–6,2)] (2) = 2(-6) + b (2) = - ½ (-6) + b 2 = - 12 + b 2 = 3 + b (4) = - 2/3(6) + b + 12 + 12 _ -3 -3 _ 14 = b -1 = b y = 2x + 14 y =-½x – 1 4= -4 + b + 4 + 4 _ 8 = b (b) Graphic Method (4) = 3/2 (6) + b 4 = 9 + b -9 -9 _ -5 = b y = - 2/3 x + 8 y = 3/2 x – 5 Itg2 U2 WS2. B EXTRA PRACTISE Integrated Math 2 – Solve by elimination WS #2-9b Solve the following system of equations by elimination 1. 6x – 3y = 12 2x + 2y = 4 (-3) 6x – 3y = 12 -6x – 9y = -12 - 12y = 0 -12 -12 y =0 (1 Step) 2x + 2(0) = 4 2x = 4 2 2 x= 2 2. 3) Write the equation of a line in slope-intercept form: (a) parallel and (b) perpendicular to the line: 3x - y = - 2 and contains the point (1, 3). Solve mathematically only - 3x + y = 2 (Make y positive by multiplying by -1) + 3x + 3x (Eventually get in the form: y = mx + b) y = 3x + 2 Parallel Slope Perpendicular Slope y = 3x + b y = - 1/3 x + b [Substitute in (1,3)] (3) = 3(1) + b 3= 3 + b -3 -3 _ 0 = b y = 3x + 0 y = 3x (3) = - 1/3(1) + b 3 = + 1/3 - 1/3 + b + 1/3 _ 3 1/3 = b y = - 1/3 x + 3 1/3 4) Write the equation of a line in slope-intercept form: (a) parallel and (b) perpendicular to the line: 2x + 3y = 9 and contains the point (6, 4). Solve mathematically only 2x + 3y = 9 (Make y positive by multiplying by -1) - 2x - 2x (Eventually get in the form: y = mx + b) 3y = - 2x + 9 3 3 y = - 2/3 x + 3 Parallel Slope Perpendicular Slope y = - 2/3 x + b y = 3/2 x + b [Substitute in (6,4)] 3x + 2y = 13 4x + 4y = 24 -6x – 4y = -26 4x + 4y = 24 - 2x = -2 -2 -2 x = 1 3(1) + 2y = 13 3 + 2y = 13 –3 –3 2y = 10 2 2 y= 5 (2,0) (-2) (1 Step) (1,5) 3. 5x + 2y = - 8 x + 7y = 5 5x + 2y = - 8 -5x – 35y = -25 -33y = -33 -33 -33 y = 1 7. (-5) (1 Step) x + 7(1) = 5 x + 7 = 5 –7 –7 x = -2 4. 6x – 5y = 31 3x + 2y = 20 6x – 5y = 31 -6x – 4y = -40 - 9y = - 9 -9 -9 y = 1 3x + 2(1) = 20 3x + 2 = 20 –2 –2 3x = 18 3 3 x =6 5. 2x + 3y = -13 5x – 6y = 8 4x + 6y = -26 5x – 6y = 8 9x = - 18 9 9 x = -2 2(-2) + 3y = -13 - 4 + 3y = -13 +4 + 4 3y = - 9 3 3 y =-3 6. 4(5) + y = 24 20 + y = 24 – 20 – 20 y = 4 (2) 4x – 2y = 2 3x + 2y = 12 7x = 14 7 7 x = 2 (1 Step) 3(2) + 2y = 12 6 + 2y = 12 –6 –6 2y = 6 2 2 y = 3 (-2,1) (-2) (2, 3) (1 Step) 8. 3x + 2y = 12 (3) 5x – 3y = 1 (2) 9x + 6y = 36 10x – 6y = 2 19x = 38 19 19 x= 2 (2 Steps) 3(2) + 2y = 12 6 + 2y = 12 –6 –6 2y = 6 2 2 y= 3 (6, 1) (2) (2, 3) (1 Step) 9. 4x – 3y = 25 -3x + 8y = 10 12x – 9y = 75 -12x + 32y = 40 23y = 115 23 23 y=5 (-2, -3) 2x – 3y = -2 4x + y = 24 (3) 2x – 3y = - 2 12x + 3y = 72 14x = 70 35 14 14 7 x= 5 2x – y = 1 3x + 2y = 12 (1 Step) (5,4) 4x – 3(5) = 25 4x – 15 = 25 + 15 + 15 4x = 40 4 4 x = 10 (3) (4) (2 Steps) (5, 10)