1. Equations of Lines MATH 018 Combined Algebra S. Rook

2. 2 Overview Sections 3.5 & 8.1 in the textbook: –Graphing Using the Slope and y-intercept –Writing Equations using the Point-Slope Formula –Equations Using Two Points –Equations of Horizontal and Vertical Lines (8.1) –Equations of Parallel and Perpendicular Lines (8.1)

3. Graphing Using the Slope and y-intercept

4. 4 Consider y = -3x + 1 There are two measurements common to any linear equation – what are they? What do we have to do in order to extract them from the linear equation? Easiest place to start is by plotting the y- intercept How can we use the slope to plot another point? –This principle is known as “rise over run”

5. Graphing Using the Slope and y-intercept (Example) Ex 1: Graph each line using the slope and y-intercept: a) y = -3x + 1 b) -2x + 3y = 6 c) y = x 5

6. Writing Equations Using the Point-Slope Formula

7. 7 Point-Slope Formula Point-Slope formula: y – y 1 = m(x – x 1 ) –(x 1, y 1 ) is any point –x and y are variables –Very similar to the slope formula discussed in section 3.4 Along with slope-intercept form, DO NOT forget about standard form: Ax + By = C where A, B, and C are constants All variables on one side and the constants on the other NO fractions

8. Point-Slope Formula (Example) Ex 2: Write the equation of the line – leave the answer in the requested form: a) Slope of 3 and passes through (-2, 4) – leave in standard form b) Slope of -½ and passes through (6, 1) – leave in slope-intercept form c) Slope of 5 and passes through (0, -9) – leave in standard form 8

9. Equations Using Two Points

10. 10 Equations Using Two Points Only difference is that we are not given the slope How do we calculate the slope between two points? We now have a choice of which point to use in the point-slope formula –It does not matter which point is used – the same equation will result

11. Equations Using Two Points (Example) Ex 3: Write the equation of the line – leave the answer in the requested form: a) Through (-1, 1) and (2, 6) – leave in standard form b) Through (2, 1) and (-1, -8) – leave in standard form 11

12. Equations of Horizontal and Vertical Lines

13. 13 Equations of Vertical and Horizontal Lines Recall the slopes of vertical and horizontal lines: –Slope of a vertical line is undefined –Slope of a horizontal line is zero Use the slope along with the given point to construct the equation of the line

14. 14 Equations of Horizontal and Vertical Lines (Example) Ex 4: Write the equation of the line: a) Slope of 0 and passes through (1, 5) b) Vertical line and passes through (0, -9)

15. Equations of Parallel and Perpendicular Lines

16. 16 Equations of Parallel and Perpendicular Lines Given the equation of a line, we want to find the equation of a second line that is parallel or perpendicular to the first Again, the slope is not explicitly given so how can we use the definitions of parallel and perpendicular lines to find the slope? Determine the appropriate slope of the second line based on whether it is to be parallel or perpendicular to the first line Use the slope and the given point in the point-slope formula

17. Equations of Parallel and Perpendicular Lines (Example) Ex 5: Write the equation of the line – leave the answer in standard form (if possible): a) Perpendicular to -4y = 3x – 8 and passes through (1, -1) b) Parallel to 16x – 8y = 5 and passes through (-3, 5) c) Perpendicular to the x-axis and passes through (-7, -11) 17

18. 18 Summary After studying these slides, you should know how to do the following: –Graph an equation given the slope and y-intercept –Write an equation given the slope and a general point –Write an equation given two points –Write equations of horizontal and vertical lines –Write equations of parallel and perpendicular lines Additional Practice –See the list of suggested problems for 3.5 Next lesson –Graphing Inequalities in Two Variables (Section 3.6)