5.1 Equations of Lines

1. 5.1 Equations of Lines y This is the graph of the equation 2x + 3y = 12. (0,4) (6,0) x 2 -2 Equations of the form ax + by = c are called linear equations in two variables. The point (0,4) is the y-intercept. The point (6,0) is the x-intercept.

2. 5.1 Equations of Lines The slope of a line is a number, m, and is defined. m is undefined positive m = 0 If a line goes up from left to right, then it has a positive slope. negative If a line goes down from left to right, then it has a negative slope.

3. 5.1 Equations of Lines y 1 1 m = 2 4 x 2 -2 m = - m is undefined The slope of a line is a number, m, which measures its steepness. m = 2 m = 0

4. y x 1 -1 5.1 Equations of Lines (-4,0) (0,-2)

5. 5.1 Equations of Lines y2 – y1 , (x1 ≠ x2). m = x2 – x1 (x2, y2) y y2–y1 change in y (x1, y1) x2–x1 change in x x The slope of the line passing through the two points (x1, y1) and (x2, y2) is given by the formula The slope is the change in y divided by the change in x as we move along the line from (x1, y1) to (x2, y2).

6. y2 – y1 5 – 3 2 1 m = = = = x2 – x1 2 4 – 2 y x 5.1 Equations of Lines Example: Find the slope of the line passing through the points (2,3) and (4,5). (4, 5) 2 (2, 3) Use the slope formula with x1= 2, y1 = 3, x2 = 4, and y2 = 5. 2

7. 5.1 Equations of Lines y = -2x + 3 x + y = 10 2x = 4 y = 5 x + 2y – 5 = 0 State the slope and the y-intercept m = -2 b = 3 m =-1 b = 10 none m = 0 b = 5 m = - 1/2 b = 5

8. 5.1 Equations of Lines A linear equation written in the form y = mx + b is in slope-intercept form. The slope is m and the y-intercept is (0, b). To graph an equation in slope-intercept form: 1. Write the equation in the form y = mx + b. Identify m and b. 2. Plot the y-intercept (0,b). 3. Starting at the y-intercept, find another point on the line using the slope. 4. Draw the line through (0, b) and the point located using the slope.

9. 5.1 Equations of Lines y x change in y 2 m = = 1 change in x (0,-4) (1, -2) Example: Graph the line y = 2x– 4. • The equation y = 2x– 4 is in the slope-intercept form. So, m = 2 and b = -4. 2. Plot the y-intercept, (0,-4). 3. The slope is 2. 2 4. Start at the point (0,4). Count 1 unit to the right and 2 units up to locate a second point on the line. 1 The point (1,-2) is also on the line. 5. Draw the line through (0,4) and (1,-2).

10. 5.1 Equations of Lines In 2000, there were 200 Twins fans at McDonell. The number will grow by 20 each year. Write a linear model and sketch the graph.

11. 5.1 Equations of Lines How many fans will there be in 5 years? When will there be 360 fans? 360 = 20t + 200 160 = 20t t = 8 years

12. 5.2 Equations of LinesPoint Slope Write an equation of the line that passes through the point (2,-1) with a slope of 2. A(2,-1)

13. 5.2 Equations of LinesPoint Slope Write an equation of the line that passes through the point (-1,-1) with a slope of -3 A(-1,-1)

14. 5.2 Equations of LinesPoint Slope Write an equation of the line that passes through the point (1,0) with a slope of 1/2 A(1,0)

15. 5.3 Equations of LinesTwo Points Write an equation of the line that passes through the point (2,1) and (3,-3) A(2,1) B(3,-3)

16. 5.3 Equations of LinesTwo Points Write an equation of the line that passes through the point (1,4) and (0, 3) A(1,4) B(0,3)

17. 5.3 Equations of LinesTwo Points Write an equation of the line that passes through the point (-1,6) and (3, -2) A(-1,6) B(3,-2)

18. 5.3 Equations of LinesTwo Points Write an equation of the line that passes through the point (1,-3) and (3, -2) A(-1,-3) B(3,-2)

19. 5.4 Exploring Data: Fitting a Line to Data

20. 5.4 Exploring Data: Fitting a Line to Data Find the equation of the line using 1928 and 1988

21. 5.4 Exploring Data: Fitting a Line to Data Find the equation of the line using 1928 and 1988

22. 5.4 Exploring Data: Fitting a Line to Data What would the time be in the year 2000?

23. 5.4 Exploring Data: Fitting a Line to Data

24. 5.4 Exploring Data: Fitting a Line to Data

25. 5.4 Exploring Data: Fitting a Line to Data Correlation: The agreement between two sets of data -1 <r< 1 r is called the correlation coefficient

26. 5.5 Standard Form of a Linear Equation

27. 5.5 Standard Form of a Linear Equation Subtract y from both sides Add 4 to both sides Switch

28. 5.5 Standard Form of a Linear Equation Multiply both sides by 2 Subtract 2y from both sides Add 2to both sides

29. 5.5 Standard Form of a Linear Equation Multiply both sides by 12 Subtract 12y from both sides Add 3to both sides

30. 5.5 Standard Form of a Linear Equation Multiply both sides by 10 Subtract 10y from both sides Add 5 to both sides

31. 5.6 Point-Slope Form of a Line 1 1 1 2 2 2 The graph of the equation y – 3 = -(x – 4) is a line of slope m = - passing through the point (4,3). y m = - 8 (4, 3) 4 x 4 8 A linear equation written in the form y–y1 = m(x – x1) is in point-slope form. It is used mainly to write the equation of a line. The graph of this equation is a line with slope m passing through the point (x1, y1).

32. Point-slope form y – y1 = m(x – x1) y – y1 = 3(x – x1) Let m = 3. y – 5 = 3(x – (-2)) Let (x1, y1)= (-2,5). y – 5 = 3(x + 2) Simplify. y = 3x + 11 Slope-intercept form 5.6 Point-Slope Form of a Line Example: Write the slope-intercept form for the equation of the line through the point (-2,5) with a slope of 3. Use the point-slope form, y – y1 = m(x – x1), with m = 3 and (x1, y1)= (-2,5).

33. m = Calculate the slope. 1 1 3 3 Point-slope form y – y1 = m(x – x1) 5 – 3 2 1 = - = - Use m = -and the point (4,3). -2 – 4 6 3 y – 3 = - (x – 4) 13 Slope-intercept form 3 y = - x + 5.6 Point-Slope Form of a Line Example: Write the slope-intercept form for the equation of the line through the points (4,3) and (-2,5).

34. 5.6 Point-Slope Form of a Line Summary of Equations of Lines Slope of a line through two points: x = a Vertical line (undefined slope): y = b Horizontal line (zero slope): y = mx + b Slope-intercept form: y2 – y1 = m(x2 – x1) Point-slope form: Ax + By = C Standard form: