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in the (x,y) plane

Dive into the world of coordinate geometry in the (x,y) plane with a focus on straight-line graphs, equation forms, gradient calculations, and practical examples to enhance your understanding and skills.

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in the (x,y) plane

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  1. Coordinate Geometry in the (x,y) plane

  2. Introduction • This Chapter focuses on coordinate geometry, mainly involving straight line graphs • We will be looking at working out equations of graphs based on various sets of information

  3. Teachings for Exercise 5A

  4. Coordinate Geometry in the (x,y) plane y Equation of a straight line The equation of a straight line is usually written in one of 2 forms. One you will have seen before; Where m is the gradient and c is the y-intercept. Or, the general form: Where a, b and c are integers. y-intercept gradient 1 x 5A

  5. Coordinate Geometry in the (x,y) plane Equation of a straight line The equation of a straight line is usually written in one of 2 forms. One you will have seen before; Where m is the gradient and c is the y-intercept. Or, the general form: Where a, b and c are integers. Example 1 Write down the gradient and y-intercept of the following graphs a)  Gradient = -3  y-intercept = (0,2) b) Rearrange to get ‘y’ on one side Divide by 2  Gradient = 2  y-intercept = (0, 5/2) 5A

  6. Coordinate Geometry in the (x,y) plane Equation of a straight line The equation of a straight line is usually written in one of 2 forms. One you will have seen before; Where m is the gradient and c is the y-intercept. Or, the general form: Where a, b and c are integers. Example 2 Write each equation in the form ax + by + c = 0 a) -y Correct form b) +1/2x and -5 x2 (to remove fraction) Correct form 5A

  7. Coordinate Geometry in the (x,y) plane Equation of a straight line The equation of a straight line is usually written in one of 2 forms. One you will have seen before; Where m is the gradient and c is the y-intercept. Or, the general form: Where a, b and c are integers. Example 3 A line is parallel to the line y = 3x + 2 and passes through (0,-1). Write the equation of the line. Parallel so the gradient will be the same Crosses through (0,-1), which is on the y-axis 2 -1 5A

  8. Coordinate Geometry in the (x,y) plane Equation of a straight line The equation of a straight line is usually written in one of 2 forms. One you will have seen before; Where m is the gradient and c is the y-intercept. Or, the general form: Where a, b and c are integers. Example 4 A line is parallel to the line 6x + 3y – 2 = 0 and passes through (0,3). Write the equation of the line. Rearrange to the form y = mx + c Divide by 3 The new line has the same gradient, but intercepts the y-axis at 3 5A

  9. Coordinate Geometry in the (x,y) plane Equation of a straight line The equation of a straight line is usually written in one of 2 forms. One you will have seen before; Where m is the gradient and c is the y-intercept. Or, the general form: Where a, b and c are integers. Example 5 The line y = 4x + 8 crosses the y-axis at 8. It crosses the x-axis at P. Work out the coordinates of P. Crosses the x-axis where y=0 -8 Divide by 4 So the line crosses the x-axis at (-2,0) 5A

  10. Teachings for Exercise 5B

  11. Coordinate Geometry in the (x,y) plane The gradient of a line You can work out the gradient of a line if you know 2 points on it. Let the first point be (x1,y1) and the second be (x2,y2). The following formula gives the gradient: ‘The change in the y values, divided by the change in the x values’ y (x2,y2) (x1,y1) y2 - y1 x2 - x1 x 5B

  12. Coordinate Geometry in the (x,y) plane The gradient of a line You can work out the gradient of a line if you know 2 points on it. Let the first point be (x1,y1) and the second be (x2,y2). The following formula gives the gradient: ‘The change in the y values, divided by the change in the x values’ Example 1 Calculate the gradient of the line which passes through (2,3) and (5,7) (x1, y1) = (2, 3) (x2, y2) = (5, 7) Substitute numbers in Work out or leave as a fraction 5B

  13. Coordinate Geometry in the (x,y) plane The gradient of a line You can work out the gradient of a line if you know 2 points on it. Let the first point be (x1,y1) and the second be (x2,y2). The following formula gives the gradient: ‘The change in the y values, divided by the change in the x values’ Example 2 Calculate the gradient of the line which passes through (-2,7) and (4,5) (x1, y1) = (-2, 7) (x2, y2) = (4, 5) Substitute numbers in Work out or leave as a fraction Simplify if possible 5B

  14. Coordinate Geometry in the (x,y) plane The gradient of a line You can work out the gradient of a line if you know 2 points on it. Let the first point be (x1,y1) and the second be (x2,y2). The following formula gives the gradient: ‘The change in the y values, divided by the change in the x values’ Example 3 Calculate the gradient of the line which passes through (2d,-5d) and (6d,3d) (x1, y1) = (2d, -5d) (x2, y2) = (6d, 3d) Substitute numbers in Work out or leave as a fraction Simplify if possible (the d’s cancel out) 5B

  15. Coordinate Geometry in the (x,y) plane The gradient of a line You can work out the gradient of a line if you know 2 points on it. Let the first point be (x1,y1) and the second be (x2,y2). The following formula gives the gradient: ‘The change in the y values, divided by the change in the x values’ Example 4 The line joining (2, -5) to (4, a) has a gradient of -1. Calculate the value of a. (x1, y1) = (2, -5) (x2, y2) = (4, a) Substitute numbers in Simplify Multiply by 2 Subtract 5 5B

  16. Teachings for Exercise 5C

  17. Coordinate Geometry in the (x,y) plane Finding the Equation of a line You can find the equation of the line with gradient m, and coordinate (x1, y1) by using the following formula: Example 1 Find the equation of the line with gradient 5 that passes through the point (3,2) (x1, y1) = (3, 2) m = 5 Substitute the numbers in Expand the bracket Add 2 5C

  18. Coordinate Geometry in the (x,y) plane Finding the Equation of a line You can find the equation of the line with gradient m, and coordinate (x1, y1) by using the following formula: Example 2 Find the equation of the line with gradient -1/2 that passes through the point (4,-6) (x1, y1) = (4, -6) m = -1/2 Substitute the numbers in Expand the brackets Subtract 6 5C

  19. Coordinate Geometry in the (x,y) plane Finding the Equation of a line You can find the equation of the line with gradient m, and coordinate (x1, y1) by using the following formula: Thought Process ‘To find the equation of the line, I need point A’ ‘Point A is on the x-axis, so will have a y-coordinate of 0’ ‘As the equation I have already, crosses A as well, I can put y=0 into it to find out the x value at A’ Example 3 The line y = 3x – 9 crosses the x-axis at coordinate A. Find the equation of the line with gradient 2/3 that passes through A. Give your answer in the form ax + by + c = 0 where a, b and c are integers. At point A, y = 0 Subtract 9 Divide by 3 A = (3,0) 5C

  20. Coordinate Geometry in the (x,y) plane Finding the Equation of a line You can find the equation of the line with gradient m, and coordinate (x1, y1) by using the following formula: Thought Process ‘To find the equation of the line, I need point A’ ‘Point A is on the x-axis, so will have a y-coordinate of 0’ ‘As the equation I have already, crosses A as well, I can put y=0 into it to find out the x value at A’ Example 3 A = (3,0) The line y = 3x – 9 crosses the x-axis at coordinate A. Find the equation of the line with gradient 2/3 that passes through A. Give your answer in the form ax + by + c = 0 where a, b and c are integers. (x1, y1) = (3, 0) m = 2/3 Substitute in values Multiply out bracket Subtract y Multiply by 3 5C

  21. Teachings for Exercise 5D

  22. Coordinate Geometry in the (x,y) plane Finding the Equation of a line You can find the equation of a line from 2 points by using the following formula: Example 1 Work out the equation of the line that goes through points (3,-1) and (5, 7). Give your answer in the form y = mx + c. (x1, y1) = (3, -1) (x2, y2) = (5, 7) Substitute in values Work out any sums Multiply the right side by 4 to make fractions the same Multiply by 8 Subtract 1 5D

  23. Coordinate Geometry in the (x,y) plane Finding the Equation of a line You can find the equation of a line from 2 points by using the following formula: Thought Process ‘We need to find point A’ ‘If the equations intersect at A, they have the same value for y (and x)’ If I can write one of the equations in terms of y, I can replace the y in the second equation and solve it’ Example 1 The lines y = 4x – 7 and 2x + 3y – 21 = 0 intersect at point A. Point B has co-ordinates (-2, 8). Find the equation of the line that passes through A and B Replace y with ‘4x - 7’ Expand the bracket Group x’s and add 42 Divide by 14 Sub x into one of the first equations to get y A = (3,5) 5D

  24. Coordinate Geometry in the (x,y) plane Finding the Equation of a line You can find the equation of a line from 2 points by using the following formula: Thought Process ‘We need to find point A’ ‘If the equations intersect at A, they have the same value for y (and x)’ If I can write one of the equations in terms of y, I can replace the y in the second equation and solve it’ Example 1 A = (3,5) The lines y = 4x – 7 and 2x + 3y – 21 = 0 intersect at point A. Point B has co-ordinates (-2, 8). Find the equation of the line that passes through A and B (x1, y1) = (3, 5) (x2, y2) = (-2, 8) Substitute in values Work out the denominators Multiply all of left by -5 and all of right by 3 (makes denominators equal) Multiply by -15 Rearrange, keeping x positive 5D

  25. Teachings for Exercise 5E

  26. Coordinate Geometry in the (x,y) plane Finding the Perpendicular to a line You need to be able to work out the gradient of a line which is Perpendicular to another.  Perpendicular means ‘intersects at a right angle… • If a line has a gradient of m, the line perpendicular has gradient -1/m • Two perpendicular lines have gradients that multiply to give -1 Example 1 Work out the gradient of the line that is perpendicular to the lines with the following gradients. Line gradient Perpendicular These lines are perpendicular 3 -1/3 1/2 -2 -2/5 5/2 2x -1/2x 5E

  27. Coordinate Geometry in the (x,y) plane Finding the Perpendicular to a line You need to be able to work out the gradient of a line which is Perpendicular to another.  Perpendicular means ‘intersects at a right angle… • If a line has a gradient of m, the line perpendicular has gradient -1/m • Two perpendicular lines have gradients that multiply to give -1 Example 2 Is the line y = 3x + 4 perpendicular to the line x + 3y – 3 = 0? Gradient = 3 These lines are perpendicular Gradient = -1/3 The lines are perpendicular since their gradients multiply to give -1 5E

  28. Coordinate Geometry in the (x,y) plane Finding the Perpendicular to a line You need to be able to work out the gradient of a line which is Perpendicular to another.  Perpendicular means ‘intersects at a right angle… • If a line has a gradient of m, the line perpendicular has gradient -1/m • Two perpendicular lines have gradients that multiply to give -1 Example 3 Find an equation for the line that passes through (3,-1) and is perpendicular to the line y = 2x - 4 Gradient = 2 These lines are perpendicular Gradient of the perpendicular = -1/2 (x1, y1) = (3, -1) m = -1/2 Substitute in values Expand brackets Subtract 1 5E

  29. Summary • We have learnt how to write equations of a line in 2 different forms • We have done this from varying sets of information • We have also looked at the link between parallel and perpendicular lines

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