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M 1313 Review Material 1 Line Formulas: Point-slope: y − y1 = m(x − x1 ) m= slope, (x1 , y1 ) is the point Slope-intercept: y = mx + b m = slope, b = y intercept General or Standard form: ax + by = c , where a, b and c are integers. Example 1: Write an equation for a line that has slope passes through the point (3. − 9 ) . 1 and 5 Example 2: Write an equation for a line which passes through the points (2 , −2 ) and (− 3 ,2 ) . Example 3: Find the slope and the y intercept for the equation 2 x − 3y = 9 M 1313 Review Material 2 Intercepts: x-intercept means y=0 y-intercept means x=0 Example 4: Find the x and y intercept of the following equation. 1 1 5 x− y= 3 7 8 Example 5: Find the equation of line that has slope of − y intercept of − 6 . 2 and a 3 Example 6: Find the equation of line that has a slope of − 3 and x-intercept of − 2 . Parallel and perpendicular lines: Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals. (This means their product is − 1 ). M 1313 Review Material 3 Example 7: Given the point (2 , −1) and the equation 2 x + 3 y = −4 a. Find equation of the line that is parallel at this point. b. Find equation of the line that is perpendicular at this point. Solutions to two equations, two unknowns: One point: No solution: All real numbers: Same Line Example 8: Find all solutions: 5x + 2 y = 2 7x + 3y = 6 M 1313 Review Material 4 Example 9: Find all solution: a. b. 2x + y = 5 4x + 2y = 8 2x + y = 4 − 6 x − 3 y = −12 Talk about the possible slopes of lines: Positive Slope: Rising right Falling left Negative slope: Rising left Falling right Example 10: Graph 5 x − 2 y = 10 y= 2, x=1 Zero or no slope Undefined slope M 1313 Review Material 5 Inequalities: < or > means the line is dashed -------≤ or ≥ means a solid line Example 11: Graph 2 x − 5 y < 20 Example 12: Graph the following inequalities and find the solution: 3x + 2 y ≥ 6 x − 3y < 9 Example 13: The solution to the linear inequality − 4 < 5 x + 3 y will be the half plane M 1313 Review Material Example 14: Find the equation of the following graph: Example 15: Find the equation of the following graph: Example 16: Find the equations of the following graph: 6 M 1313 Review Material Example 17: I. II. 7 Which of following are true? 7 3 The point , is a point on the graph 10 x − 2 y > 6 . 10 16 L1 : − 2 x + 5 y = 5 and L2 : 4 x − 10 y = −4 are perpendicular lines. III. A line through the points (− 4 ,2 ) and (− 4 ,7 ) is a horizontal line. IV. (1,−3) is in the solution set x > 0 and y ≤ 0 . Example 18: Which of the following are not true? I. II. y = 2 x − 5 has a slope that falls to the right. A horizontal line has a slope of zero. III. A line with negative slope falls to the left.