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Math 99 – Test 1 Review Spring 2013 Math 99 – Test 1 Practice 1. For the function f(x) = x2 – 2 x + 1 Find the following (a) f(2) (b) f(– 3) (c) f( ½ ) 2. Solve the equation 3x + 7 = 28 3. Solve the equation 5x + 7 = – 2x + 35 4. Solve the equation 3(2x – 2) 5. Solve the equation 5 = 24 10 x 6. Solve the equation x 4 2 = 7. Solve the following Inequalities. (a) 5x + 5 > 3x – 9 show all your working. show all your working. = show all your working. 4x – 4 show all your working. show all your working. 1 x2 3 show all your working. (b) 2(x – 7) + 9 < 7x + 15 (c) 8. Change the formula y = mx + b to be in terms of m 9. Peter’s height is 5 inches more than three times Mary’s height. If Mary height is m inches write down Peters height in terms of m. 10. A bookcase is to have five shelves as shown in the diagram below. The height of the bookcase is to be two feet more than its width. Find the width and height of the bookcase if 50 feet of lumber was used in the construction of the bookcase. 11. Express each of the system of equations below in the slope intercept form ( y = ….. form). And without drawing graphs identify each system of linear equations below as consistent, inconsistent or dependant. Also indicate the number of solutions that would occur. (a) y 2y = = 2x + 4 4 – 2x (b) x – 2y 3x = = 10 6y + 15 (c) x–4 6y + 8 = = 3y 2x Page 1 Math 99 – Test 1 Review Spring 2013 12. Determine which of the following points, A(2,1) , B(1, – 3) and C(– 1 , 3) if any, satisfy both pairs of equations. y 3x – 2y = = 4x – 7 4 13. Graph the line 3x + 2y = 6 by plotting at least 3 points. 14. What is the slope and y-intercept of the line with equation 4x + 2y = 20 15.(a) Graph the following lines. 2x + 3y 4x – 2y = = 15.(b) From the graph what is the solution to this system of equations. 16.(a) Solve the system of equations y y 16.(b) Solve the system of equations 16.(c) 12 8 3x – 1 5x – 5 by using the substitution method. 2x + y 3x + 4y = 1 = –1 by using the substitution method. Solve the system of equations 5x – 2y 5 = = –7 y – 3x by using the substitution method. 16.(d) Solve the system of equations 4x – 2y y = = –7 2x – 1 by using the substitution method. 17.(a) Solve the system of equations 5x – 2y 2x + 4y = = 8 8 by using the addition method. 17.(b) Solve the system of equations 2x – 2y 4x + 3y = = 10 –1 by using the addition method. 17.(c) Solve the system of equations = = –1 y–2 by using the addition method. 4x 18. = = A truck rental agency charges a daily fee plus a mileage fee. Julie was charged $85 for two days and 100 miles and Christina was charged $165 for 3 days and 400 miles. What is the agency’s daily fee and what is the mileage fee? 19. The plumber charges a fixed rate for turning up at your house plus a charge per hour for the work done. From previous jobs that the plumber has done it was found that a 4 hour job will cost a total of $283 while a 6 hour job will cost a total of $387.What is the fixed rate and the charge per hour? 20. By weight one alloy of brass is 70% Copper and 30% Zinc. Another Alloy is 40% Copper and 60% Zinc. How many grams of each alloy would need to be melted and combined to obtain 600 grams of a brass alloy that is 60% Copper and 40% Zinc? Page 2 Math 99 – Test 1 Review Spring 2013 Math 99 – Test 1 Practice Solutions 1. For the function f(x) = x2 – 2 x + 1 Find the following (a) f(2) = (2)2 – 2 (2) + 1 = 4 – 4 + 1 = 1 (b) f(– 3) = (– 3)2 – 2 (– 3) + 1 = 9 + 6 + 1 = 16 (c) f( ½ ) = (½)2 – 2 (½) + 1 = ¼ – 1 + 1 = ¼ 2. 3. Solve the equation 3x + 7 = 28 3x + 7 3x x subtract 21 from both sides divide both sides by 3 = = = = – 2x + 35 + 35 28 4 Solve the equation 3(2x – 2) 6x – 6 2x – 6 2x x 5. 28 21 7 Solve the equation 5x + 7 = 5x + 7 7x + 7 7x x 4. = = = = = = = = 10 x 5x = x = – 240 – 48 – 2x + 35 show all your working. add 2x to both sides subtract 7 from both sides divide both sides by 7 3(2x – 2) = 4x – 4 4x – 4 –4 2 1 Solve the equation 5 = 24 show all your working. 5 = 24 4x – 4 show all your working. multiply out 3(2x – 2) subtract 4x from both sides add 6 to both sides divide both sides by 2 10 x show all your working. cross multiply divide both sides by 5 Page 3 Math 99 – Test 1 Review Spring 2013 6. Solve the equation x 4 = 2 1 x4 = 2 1 1 x x4 2 3 1 x4 6 1 x 6 x 7. x 4 = 2 1 x2 3 1 x2 3 = –2 = –2 = –6 = – 36 1 x2 3 show all your working. 1 x as being the same as x 2 2 1 subtract x from both sides 3 1 1 1 simplifying x x = x 2 6 3 think of subtract 4 from both sides multiply both sides by 6 or divide both sides by 1 6 Solve the following Inequalities. (a) 5x + 5 5x 2x x > > > > 3x – 9 3x – 14 – 14 –7 (b) 2(x – 7) + 9 2x – 14 + 9 2x – 5 2x – 5x x < < < < < > 7x + 15 7x + 15 7x + 15 7x + 20 20 –4 (c) ( multiply both sides by 12 ) (divide both sides by – 3 and switch the inequality symbol ) 8. Change the formula y y–b yb x 9. y = mx + b to be in terms of m = = mx + b mx subtract b from both sides = m divide both sides by x Peter’s height is 5 inches more than three times Mary’s height. If Mary height is m inches write down Peters height in terms of m. Peter’s height = 3 times Mary’s height + 5 = 3m + 5 Page 4 Math 99 – Test 1 Review Spring 2013 10. A bookcase is to have five shelves as shown in the diagram below. The height of the bookcase is to be two feet more than its width. Find the width and height of the bookcase if 50 feet of lumber was used in the construction ww of the bookcase. Total amount of wood w + 2 + w +w + w + w +w + w + 2 7w + 4 7w 50 feet w+2 50 50 46 46 w = or 6.57 feet 7 width of bookshelf = w = 6.75 feet and height of bookshelf = 6.75 + 2 w w w = = = = w+2 w w w w = 8.75 feet 11. Express each of the system of equations below in the slope intercept form ( y = ….. form). And without drawing graphs identify each system of linear equations below as consistent, inconsistent or dependant. Also indicate the number of solutions that would occur. (b) y 2y = 2x + 4 = 4 – 2x 2y y = = 4 – 2x 2–x y = –x+2 The two equations are Now (b) x – 2y 3x (c) x–4 6y – 8 = = 10 6y + 15 x – 2y – 2y = = 10 – x + 10 x–4 = 3y 3y = x – 4 y = ½x–5 y 3x 6y + 15 = = 6y + 15 3x 6y + 8 6y 6y = 3x – 15 y y = ½x – 2½ = = 3y = 2x 1 4 x 3 3 = = 2x 2x –8 y y = = 2x + 4 –x+2 Consistent # of solutions = 1 Inconsistent # of solutions = 0 = 1 4 x 3 3 Dependant # of solutions = Page 5 Math 99 – Test 1 Review Spring 2013 12. Determine which of the following points, A(2,1), B(1, – 3) and C(– 1 , 3) if any, satisfy both pairs of equations. y 3x – 2y = = 4x – 7 4 Solution: Test each point separately A(2,1) this means x = 2 and y = 1 In equation y 1 1 = = = 4x – 7 4(2) – 7 1 3x – 2y 3(2) – 2(1) 4 (works) = = = 4 4 4 (works) = = = 4 4 4 (Fails) Answer = A(2,1) is on both lines and so satisfies both equations. Test each point separately B(1, – 3) this means x = 1 and y = – 3 In equation y = –3 = –3 = 4x – 7 4(1) – 7 –3 3x – 2y 3(1) – 2(– 3) 9 (works) Answer = B(1, – 3) is not on both lines and so does not satisfy both equations. Test each point separately C(– 1,3) this means x = – 1 and y = 3 In equation y 3 3 = = = 4x – 7 4(–1) – 7 – 11 (Fails) Answer = C(– 1,3) is not on the first line and so does not satisfy both equations. 13. Graph the line 3x + 2y = 6 by plotting at least 3 points. y x y 0 3 2 0 4 –3 x 14. What is the slope and y-intercept of the line with equation 4x + 2y = 20 4x + 2y 2y y = = = 20 – 4x + 20 – 2x + 10 Slope = m = – 2 y -intercept is (0,10) Page 6 Math 99 – Test 1 Review Spring 2013 15. (a) On the grid opposite graph the following lines 2x + 3y 4x – 2y = = y 12 8 Working : Table of values : x=0 x=3 x=6 0 For 4x – 2y = 8 choose For 2x + 3y choose y=4 y=2 y=0 y=–4 y=–2 y=0 x=0 x=1 x=2 15.(b) From the graph above what is the solution to this system of equations. Solution is the point (3,2) 16.(a) Solve the system of equations y Substitute y = 3x – 1 into the equation y = = 3x – 1 5x – 5 y = 3x – 1 3x – 2x x Substitute x = 2 into y = 3x – 1 = 3(2) – 1 = 6 – 1 = 5 16.(b) Solve the system of equations Rearrange the equation 2x + y Substitute y = = = = = by using the substitution method. 5x – 5 = 5x – 5 = 5x – 4 = –4 = 2 add 1 to both sides subtract 5x from both sides divide both sides by – 2 So the Solution is the point (2,5) 2x + y = 1 by using the substitution method. 3x + 4y = – 1 1 so that it is in the form y = ....... so 2x + y = 1 becomes y = 1 – 2x = 1 – 2x into the equation Substitute x = 1 into 2x + y 2(1) + y 2+y y x 1 1 1 –1 3x + 4y 3x + 4(1 – 2x) 3x + 4 – 8x – 5x + 4 – 5x x = –1 = –1 = –1 = –1 = –5 = 1 So the Solution is the point (1,– 1 ) Page 7 Math 99 – Test 1 Review Spring 2013 16.(c) Solve the system of equations Re arrange 5 = y – 3x Substitute y = 3x + 5 into to get 5x – 2y 5 = = –7 y – 3x 5 y – 3x y = = = by using the substitution method. y – 3x 5 3x + 5 5x – 2y 5x – 2(3x + 5) 5x – 6x – 10 – x – 10 –x x = = = = = = –7 –7 –7 –7 3 –3 Use x = – 3 in equation y = 3x + 5 = 3(– 3) + 5 = – 9 + 5 = – 4 The solution to the above system of equations is 16.(d) Solve the system of equations (– 3, – 4) 4x – 2y y Substitute y = 2x – 1 into the first equation = = –7 2x – 1 4x – 2y 4x – 2(2x – 1) 4x – 4x +1 1 = = = = by using the substitution method. –7 –7 –7 –7 Since 1 = – 7 is wrong we can conclude that this system of equations will have no solutions. This would happen when the two lines are Parallel. 17.(a) Solve the system of equations Solution: 5x – 2y = 2x + 4y = 8 8 5x – 2y 2x + 4y = = 8 8 equation1….. multiply by 2 equation 2 ….leave alone Substitute x = 2 into equation 2x + 4y 2(2) + 4y 4 + 4y 4y y = = = = = by using the addition method. 10x – 4y 2x + 4y 12x x = = = = 16 8 24 2 add equations 8 8 8 4 1 So the Solution is the point (2,1) Page 8 Math 99 – Test 1 Review Spring 2013 17.(b) Solve the system of equations 2x – 2y 4x + 3y = = 10 1 2x – 2y 4x + 3y = multiply by – 2 put y = – 3 into equation = 10 –1 – 4x + 4y 4x + 3y 7y y 4x + 3y 4x + 3( – 3) 4x – 9 4x x = = = = = –1 –1 –1 8 2 = = –1 y–2 by using the addition method. = = = = – 20 –1 – 21 –3 add the equations So the Solution is the point (2, – 3 ) 17.(c) Solve the system of equations 4x 4x = 5x – 2y 4x – y = –1 y–2 = = – 10 –2 Put x = 2 into equation by using the addition method. multiply both sides by 10 subtract y from both sides 5x – 2y 4x – y = = – 10 –2 multiply both sides by – 2 Add the equations 5x – 2y – 8x + 2y – 3x x = = = = – 10 4 –6 2 4x 4(2) 8 10 = = = = y–2 y–2 y–2 y Solution is (2,10) 18. A truck rental agency charges a daily fee plus a mileage fee. Julie was charged $85 for two days and 100 miles and Christina was charged $165 for 3 days and 400 miles. What is the agency’s daily fee and what is the mileage fee? Julie was charged $85 for two days and 100 miles Christina was charged $165 for 3 days and 400 miles 2d + 100m 3d + 400m = = 85 165 Put m = 0.15 into the equation multiply by – 3 multiply by 2 Add the equations 2d + 100m 3d + 400m = = 85 165 – 6d – 300m 6d + 800m 500m m = = = = – 255 330 75 0.15 2d + 100m = 2d + 15 2d d 85 = = = 85 70 35 So solution is d = $35 per day and m = $0.15 per mile Page 9 Math 99 – Test 1 Review Spring 2013 19. The plumber charges a fixed rate for turning up at your house plus a charge per hour for the work done From previous jobs that the plumber has done it was found that a 4 hour job will cost a total of $283 while a 6 hour job will cost a total of $387. What is the fixed rate and the charge per hour? Let x y = = fixed amount charge per hour From the information a 4 hour job will cost a total of $283 we get the equation x + 4y From the information a 6 hour job will cost a total of $387 we get the equation x + 6y Use the substitution method so we rearrange to get Use x = 283 – 4y into equation x + 4y x x + 6y 283 – 4y + 6y 283 + 2y 2y y = = = = = = = = = 283 387 283 283 – 4y 387 387 387 104 52 Use y = 52 in equation x = 283 – 4y = 283 – 4(52) = 283 – 208 = 75 So the plumber had a fixed amount of $75 plus he charged $52 per hour. 20. By weight one alloy of brass is 70% Copper and 30% Zinc. Another Alloy is 40% Copper and 60% Zinc. How many grams of each alloy would need to be melted and combined to obtain 600 grams of a brass alloy that is 60% Copper and 40% Zinc? x y = = amount of alloy one amount of alloy two x+y 0.7x + 0.4y = = (70% Copper and 30% Zinc) (40% Copper and 60% Zinc) 600 0.6(600) = Rearrange equation 1: 0.7x + 0.4 (– x + 600) 0.7x – 0.4x + 240 0.3x + 240 0.3x x and y 360 to get = = = = = = Equation 1: Equation 2: y = – x + 600 and substitute this into equation 2: 360 360 360 120 400 grams 200 grams So we mix 400 grams of alloy one with 200 grams of alloy 2. Page 10