5. In the Cartesian plane, a line runs through the points (5, 6) and (-2, -2). What is the slope of the line?
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1 Slope review Using two points to find the slope In mathematics, the slope of a line is often called m. We can find the slope if we have two points on the line. We'll call the first point and the second point. To find the slope we make a fraction. Take the y value from the second point and subtract the y value from the first point for the numerator and take the x value from the second point and subtract the x value from the first point for the denominator. slope = m = = = The slope of the line that contains the points A (2,3) and B (0,-1) is = = = 2 The slope of the line that contains the points A (-2,3) and B (0,-1) is = = = What is the slope between (1,0) and (5,3)? 2. What is the slope between (3,0) and (1,2)? 3. What is the slope between the points (-1, 1) and (3,1)? 4. In the coordinate plane a line runs through the points (2,3) and (4, 5). What is the slope of the line? 5. In the Cartesian plane, a line runs through the points (5, 6) and (-2, -2). What is the slope of the line? 1
2 Finding Slope and y-intercept from Slope-Intercept Form Slope-intercept form is y = mx + b 6. What is the slope of the line y = X - 7? 7. At what point does the line y = X + 3 cross the y axis? 8. What is the slope of the line y = X - 4? 9. What is the slope of the line y = -X - 2? 10. Which of the following is an equation for the graph in the following standard (x,y) coordinate plane? A. y = -2x + 1 B. y = x + 1 C. y = x + 2 D. y = 2x + 1 E. y = 2x What is the y-intercept of the line described by the equation y = x - 12? 2
3 Re-writing equations in slope-intercept form To find the slope or y-intercept of a line that is not in slope-intercept form, re-write the equation in slope-intercept form. For example, to find the slope of the line 6y - 3x = 18, re-write it so it looks like y = mx + b. 12. At which y-coordinate does the line described by the equation 6y - 3x = 18 intersect the y-axis? 13. What is the slope of the line described by the equation y - 2x = 8? 14. What is the slope of the line represented by the equation 10y - 16x = 13? 15. What is the slope of the line given by the equation 8 = 3y - 5x? 16. The formula for line q in standard form is 5x - y = 2. Which of the following gives the formula for line q in slope-intercept form? A. y = 5x + 2 B. y = 5x - 2 C. y = 2x - 5 D. y = -5x - 2 E. y = -5x + 2 3
4 Creating an equation for a line from 2 points Finding an equation for a line when we are only given 2 points takes 2 steps. We know that y = mx + b, where m is the slope and b is the y -intercept. So in order to create the equation we need m and we need b. Step 1: find m (slope) We already know how to find the slope from 2 points. slope = m = = = Step 2: find b (y-intercept) We're going to use the formula y = mx + b to solve for b. Plug in the slope you found in step 1 for m, then choose one of the points that was given (it doesn't matter which one), plug in the x value for x in the formula and the y value in for y in the formula. Now the only variable that doesn't have a known value is b. Solve for b. Example: What is the equation of the line that passes through the points A (2,3) and B (0,-1)? Step 1: find m (slope) slope = m = = = = = = 2 Step 2: find b (y-intercept) y = mx + b (start with the generic slope-intercept formula) 3 = 2(2) + b [plugging in the slope (2) for m and point (2,3) for x and y in the equation] 3 = 4 + b (multiply) -1 = b (subtract 4 from each side) y = mx + b (now take the generic formula for a line and... y = 2x -1 plug in the values you found for m and b) 4
5 17. What is the equation of the line that passes through points A (-2,3) and B (0,-1)? 18. What is the equation of the line that passes through the points (1,0) and (5,3)? 19. What is the equation of the line that passes through the points (3,0) and (1,2)? 20. In the coordinate plane a line runs through the points (2,3) and (4, 5). What is the equation of the line? 21. In the Cartesian plane, a line runs through the points (5, 6) and (-2, -2). What is the equation of the line? 5
6 Parallel lines on the coordinate plane Parallel lines always have the same slope. Therefore, if two lines have the same slope then they are parallel. 22. These 2 lines are parallel: y = 2x + 3 y = 2x - 4 True False 23. These 2 lines are parallel: y = x + 3 y = x True False 24. These 2 lines are parallel: y = x + 5 y = x - 3 True False 6
7 25. These 2 lines are parallel: x + 2y = 3 2y = x - 7 True False 26. These 2 lines are parallel: 3y - x = 6 3y = 2x +9 True False 27. Which of these is an equation of a line with a y-intercept of 3 and is parallel to the line 3y - x = 6 A. y = x + 2 B. y = x + 3 C. y = 3x + 2 D. y = 2x + 6 E. y = x Which of these is an equation of a line parallel to the line 2y = x - 7 which goes through the point (0, 5)? A. y = 5x - 7 B. y = 2x + 5 C. y = x - 7 D. y = x + 5 E. y = x Write the equation of a line which is parallel to the line y = x + 3 and passes through the point (0, -1) 7
8 Finding the distance between two points To find the distance between two points, use the Pythagorean theorem. The difference between is one leg and the difference between and is the other leg. and For example, in the figure below we have two points (-2,2) and (1,-2). To find the distance between them, we make a right triangle. The difference between (which is -2) and (which is 1) equals 3, so 3 is the length of the base leg. The difference between (which is 2) and (which is -2) equals 4, so the length of the other leg is 4. Now we have two sides of a right triangle and we can use the Pythagorean theorem to find the length of the hypotenuse. + = + = = 5 = c So the distance between point (-2,2) and (1,-2) is 5. If you prefer, rather than work through the Pythagorean theorem, you may memorize the distance formula which is just the points plugged into the Pythagorean theorem. 30. What is the distance between the points (3,6) and (5,-2)? A. 68 B. 10 C. 8 D. 2 E. 2 8
9 31. How many units apart are the points P (-1, -2) and Q (2,2) in the standard (x,y) coordinate plane? 32. Point M (2,3) and point N (6,5) are points on the coordinate plane. What is the length of the segment MN? 33. Points A and B lie in the standard (x,y) coordinate plane. The (x,y) coordinates of A are (2,1), and the (x,y) coordinates of B are (-2,-2). What is the distance from A to B? 34. What is the distance, in coordinate units, between points J (-5,4) and K (6,-2) in the standard (x,y) coordinate plane? 35. Point A (-2,9) and point B (3, -3) are points on the coordinate plane. What is the length of the segment AB? 36. On a sonar map in the standard (x,y) coordinate plane, the Yellow Submarine and the Sandwhich Submarine are located at the points (-7,4) and (-2,6), respectively. each unit on the map represents an actual distance of 5 nautical miles. Which of the following is closest to the distance, in nautical miles, between the two submarines? A. 5 B. 19 C. 27 D. 30 E. 67 9
10 Answers: E B 17. y = -2x y = x y = -x y = x y = x True 23. True 24. False 25. False 26. False 27. B 28. D 29. y = x D or C 10
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