1. Linear Equations in Two Variables MATH 109 - Precalculus S. Rook

2. Overview Section 1.3 in the textbook: – Graphing a linear equation by using its slope – Finding the slope of a line – Writing Linear Equations – Parallel and Perpendicular Lines 2

3. Graphing a Linear Equation by Using its Slope

4. 4 Graphing Lines in Slope-Intercept Form Slope-Intercept Form of a line: y = mx + b where m is the slope and the y-intercept is (0, b) To graph a line using its slope and y-intercept: – Solve the equation for y if necessary – Plot the y-intercept (0, b) – Use rise over run with the slope to get 2 or 3 more points

5. Other Lines Two other type of lines: – When a linear equation is only in 1 variable (x or y) – Vertical lines have the form x = a (a is a constant): m = Ø (undefined) To sketch, find x = a on the x-axis and draw a vertical line – Horizontal lines have the form y = b (b is a constant) m = 0 To sketch, find y = b on the y-axis and draw a horizontal line 5

6. Graphing a Linear Equation by Using its Slope (Example) Ex 1: Sketch the graph of the linear equation using its slope and y-intercept: a) b) 6

7. Graphing Other Lines (Example) Ex 2: Sketch the line: a)y + 3 = 2 b)x = 4 7

8. Finding the Slope of a Line

9. 9 Definition and Properties of Slope Slope (m): the ratio of the change in y (Δ y) and the change in x (Δ x) – Quantifies (puts a numerical value on) the “steepness” of a line Given 2 points on a line, we can find its slope:

10. 10 Sign of the Slope of a Line To determine the sign of the slope, examine the graph of the line from left to right: – Positive if the line rises – Negative if the line drops

11. Finding the Slope of a Line (Example) Ex 3: Find the slope of the line: a)Through (-2, 5) and (4, -5) b)Vertical line through (3, 11) c)Through (0, 8) and (1, 10) 11

12. Writing Linear Equations

13. 13 Point Slope Formula & Standard Form of a Line Given the slope and a point on a line, we can construct its equation 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 – Could also use y = mx + b Standard form: a linear equation in the form Ax + By = C where A, B, and C are constants – Variables to the left and constants on the right – NO fractions or decimals

14. Writing an Equation in Standard Form Ex 4: Write the linear equation of the line in standard form: a) Through (-4, -8) with a slope of -¼ b)Through (2, 0) with a slope of 1 14

15. 15 Equations of Lines when Given Two Points We know how to find the slope of a line given two points Proceed as before – Pick one of the two points to use in the point- slope formula

16. Writing the Equation of a Line Given Two Points (Example) Ex 5: Write the equation of the line in slope- intercept form: a) Through (4, 7) and (8, 9) b)Through (2, 1) and (-3, 6) 16

17. 17 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: – If the line is vertical, use the x-coordinate of the given point – If the line is horizontal, use the y-coordinate of the given point

18. Equations of Vertical and Horizontal Lines (Example) Ex 6: Write the equation of the line: a) Through (-2, 5) with undefined slope b)Through (4, 13) with a slope of zero 18

19. Parallel and Perpendicular Lines

20. 20 Slopes of Parallel and Perpendicular Lines Parallel lines: two lines that have the SAME slope Perpendicular lines: two lines that have OPPOSITE RECIPROCAL slopes – i.e. the product of the slopes is -1

21. Classifying Lines as Parallel, Perpendicular, or Neither (Example) Ex 7: Determine whether the two lines are parallel, perpendicular, or neither: a)3x + y = -4 and 6y – 2x = 12 b)y = 2x + 1 and y = -2x – 3 c)3y – 15x = -6 and y = 5x + 2 21

22. 22 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 Slope is not explicitly given – Put the equation of the given line in slope-intercept form Determine the appropriate slope based on whether the second line is to be parallel or perpendicular to the first Use the slope and the given point in the point-slope formula or y = mx + b A vertical line is parallel to another vertical line and a horizontal line is perpendicular to a vertical line – Vice versa for a horizontal line

23. Equations of Parallel and Perpendicular Lines (Example) Ex 8: Write the equation of the line in standard form if possible: a)Perpendicular to y = -3 and passes through (1, -5) b)Parallel to y = ½x + 5 and passes through (4, 2) c)Perpendicular to 3x + y = -1 and passes through (-1, -1) 23

24. Summary After studying these slides, you should be able to: – Graph a line using its slope and y-intercept – Graph vertical or horizontal lines – Find the slope of a line – Write linear equations – Classify two lines as being parallel, perpendicular, or neither – Write equations involving parallel and perpendicular lines Additional Practice – See the list of suggested problems for 1.3 Next lesson – Functions (Section 1.4) 24