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Practice Problems 2 Lines and Slope Part I. Forms of a linear relation. a) Standard form or general form: Ax + By = C or Ax + By + C = 0 b) Slope-intercept form: y = mx + b c) Intercept-form x y 1 where a= the x-intercept and b=the y=intercept d) a b e) Point-slope form: y y1 m( x x1 ) f) Vertical line: x = h g) Horizontal line: y = k Find the slope of the following lines 1. The line that contains the points (0, 0) and (2, 1). 2. The line containing the points (-2, 2), (1, 1). 3. The line containing the points (2, 3), (4, 0). 4. The line containing the points (-2, 3), (2, 1). 5. The line containing the points (-3, -1), (2, -1). 6. The line containing the points (-1, 2), (-1, -2). -1- Graph by hand (no calculator) 1. The line containing the point P=(1, 2) with slope m=3 y x Find the equation of the following lines 1. Containing the point P=(1, 2), with slope m=4 2. P=(2, -4), m 3 2 3. The line containing the points (0, 0) and (2, 1) 4. The line containing the points (-1, 3) and (1, 1) 5. Slope =3; containing the point (-2, 3) -2- 6. Containing the points (1, 3) and (-1, 2) 7. slope m=-3 and y-intercept=3 8. x-intercept = 2, y-intercept = -1 Therefore the line contains the points (2, 0) and (0, -1) 9. Parallel to y=2x; containing the point (-1, 3) 10. Parallel to the line 2x – y = -2; containing the point (0, 0) 11. Parallel to the line x = 5 and containing the point (-1, 2). 12. Perpendicular to the line y 1 x 4 ; containing the point (1, -2) 2 -3- 13. Perpendicular to the line x = 8; containing the point (3. 4) 14. Find the equation of the line perpendicular to the line 2x+y=2; containing the point (-3, 0) 15. Find the equation of the line perpendicular to the line x - 2y=-5; containing the point (0, 4) 16. A car cost $20,000 and depreciates linearly at the rate of $3000 per year. Find the equation of linear depreciation and the value of the car 5 years from now. Solution: 17. Problem: Peter bought a car for $18000 knowing that it will depreciate to $3000 in 5 years. a) What is the rate of linear depreciation? a) ___________ b) What is the equation of the linear depreciation? c) In how many years will the car have no value? -4- b) ________________ c) ____________ 18. The Centigrade scale and the Fahrenheit scale are linearly related. Water freezes at which is the same as 32 F and boils at 100 C which is the same as 212 F . a) Find the linear equation that give F as a function of C 0 C a) ____________ b) Find the linear equation that give C as a function of F c) Convert 100 F to C b) ____________ c) ____________ d) Convert 40 C to F d) ____________ 19. A four year old car has a value of $18000 and in 6 more years the value will be $600. If the car depreciated linearly, what was the original price of the car? Solution: 20 - Graph the region intersection of the inequalities 2x – 3y < 5 and 3x +4y >4 -5- Part II. Equation y=mx +b Slope m y-intercept b 3 x5 5 1) y 2) f(x)= -2 x +3 y= -2x + 3 f ( x) 5 3) 1 x 2 1 y x5 2 5) 3x + 2y = 10 6) 2x – 3y =12 7) 4x – 3f(x) –15=6 or 4x – 3y =21 8) 9 = 3 + 5x –2f(x) or 5x – 2y =6 9) m 2 , y int ercept (0, 4) 9 Find the equation of the line. 11) m 4, y int ercept (0, 7) Find the equation of the line 3 11) m 4.2 y int e rcept (0, ) 4 Find the equation of the line 12 ) m 2 , passe sthrough (3, 7 ). Use the po int slope form 9 Find the equation of the line -6- 3 13) m , passes through ( 4, 1) 5 Find the equation of the line . Use the po int slope form 14. The line contains the points (-1, 5) and (2, -4) Find the equation of the line Use the po int slope form 15. The line contains the points (7, 0) and (-1, 4) Find the equation of the line . Are the given lines parallel, perpendicular or neither? x 2y 5 16. 2x 4y 8 Are the given lines parallel, perpendicular or neither? y 4x 5 17. 4y 8 x 2 x 1 7 2 Parallel line : m , po int (3, 5) 7 Po int slope form : y y 1 mx x1 18. (3, 5), y 2 2 29 ( x 3) 7 y 35 2x 6 7y 2x 29 y x 7 7 7 7 Perpendicu lar line : m , po int (3, 5) 2 Po int slope form : y y 1 mx x1 y5 7 7 31 y 5 ( x 3) 2y 10 7x 21 2y 7 x 31 y x 2 2 2 -7- 19. Given a line with point P=(1, 2), and slope m=4, determine three other points of the line 20. Determine the equation of a line perpendicular to the line x = 8; containing the point (3. 4) 21. Given a line with point P=(2, -4), and slope m 3 , determine three other points of the 2 line 22. Determine the equation of the line thaqt has the points (0, 0) and (2, 1) 23. Determine the equation of the line parallel to the line 2x – y = -2; containing the point (0, 0) The line 2x – y = -2 has slope m = 2. The line we are looking for has the same slope and contains the point (0, 0) 24. Determine the equation of the line parallel to the line x = 5 and containing the point (-1, 2). The line x = 5 is a vertical line with no slope. 25. Determine the equation of the line perpendicular to the line y (1, -2) -8- 1 x 4 ; containing the point 2 1 1 x 4 has slope m . The slope of a perpendicular line is the negative 2 2 reciprocal and therefore has slope m = -2. Use the po int slope form The line y y y1 m( x x1 ) y 2 2( x 1) y 2 2 x 2 y 2 x is the slope int ercept form or 2 x y 0 is the s tan dard or general form -9-