Math 1313 Section 1.2: Straight Lines In this section, we ll review plotting points, slope of a line and different forms of an equation of a line. Graphing Points and Regions Here s the coordinate plane: As we see the plane consists of two perpendicular lines, the x-axis and the y-axis. These two lines separate them into four regions, or quadrants. The pair, (x, y), is called an ordered pair. It corresponds to a single unique point in the coordinate plane. The first number is called the x coordinate, and the second number is called the y coordinate. The ordered pair (0, 0) is referred to as the origin. The x coordinate tells us the horizontal distance a point is from the origin. The y coordinate tells us the vertical distance a point is from the origin. You ll move right or up for positive coordinates and left or down for negative coordinates.
Example: Plot the following points. A. (-2,6) B. (3,-4) C. (5,3) D. (-7,-3) E. (4,-5) F. (-1,7)
Slope of a Line If (x1, y1) and (x2, y2) are any two distinct points on a non vertical line L, then the slope m of L is given by = = When the m = 0, you have a horizontal line. When the m = undefined, you have a vertical line. Example: Find the slope between the points. a.(4, 8) and ( 3,6) b.(1,4) and ( 3,4)
c.( 1, 7) and ( 1,12) Equations of Lines Every Straight line in the xy-plane can be represented by an equation involving the variables x and y. The first from we will be looking at Point -Slope Form An equation of the line that has the slope m and passes through the point (x 1, y 1) is given by = ( ) Example: Find the equation of the line that pass through (4,7) and (-4,-9) Example: Write the equation of a line that has slope -4/3 and passes through (6, -8/3) Note: Parallel lines are two distinct lines that have equal slopes or both are undefined. Perpendicular Lines are two distinct lines that have slopes m 1 and m 2 and they are negative reciprocals of each other. m 1 *m 2 =-1
Example: Give the equation of the line that is parallel to y=-3x+4 and passes through (2, -4) Example: Give the equation of the line that is perpendicular to = +1 and passes through (-6,4) Slope Intercept Form When an equation is left in the form of = +, where m is the slope and b is the is the y-intercept of the line. General Equation of a Line is in the form + + = 0, w
Math 1313 Section 3.1: Graphing Systems of Linear Inequalities in Two Variables In this section, we ll review graphing some inequalities. Linear inequalities are in the form of: + + <0 + + >0 + + 0 + + 0 Procedures for graphing inequalities: 1. Draw the line of the inequality replacing < or > with =, if its < or > the line you draw will be dashed not solid. 2. Pick a test point on either side of the line and plug it into the original inequality 3. If the point picked works then that s the side you shade in. If it is not true, shade the other side. Example: Determine the solution set for 2x+4y >12
Example: Determine the solution set for 3 6 12 Example: Determine the solution set for 3 +2 <4 and 2 +4 > 8
Math 1313 Prerequisites Equations (These will not be given on the test. You need to memorize them.) y2 y1 Slope of a line: m = x2 x1 Slope-Intercept Form: y = mx + b Point-Slope Form: y y1 = m( x x1) Standard Form: ax + by = c Example 1: Write an equation for the line that has slope 5 1 and passes through (3, -9). Example 2: Write an equation for the line that passes through (-2, 5) and (4, 8). Math 1313 - Prerequisites 1
Parallel and Perpendicular Lines Two lines are parallel if and only if their slopes are the same. Two lines are perpendicular if and only if their slopes are negative reciprocals of each other. Example 3: Find an equation of the line that passes through the point (-2, 2) and is parallel to the line 2x 4y 8 = 0. Example 4: Find an equation of the line that passes through the point (-1, -3) and is perpendicular to the line that passes through (3, -4) and (9, -6). Example 5: Find the point of intersection. 2x 5y = 2 a. x 4y = 2 b. 2 y = 4 x Math 1313 - Prerequisites 2
3 x + 6y = 13 c. 2 x y = 3 2y = 4x 6 Example 6: Graph the following system of inequalities. x y 6 a. x + y 1 Math 1313 - Prerequisites 3
b. 3x + 2y > 6 x 3y < 9 x 0 Example 7: Write a system of linear inequalities that describes the shaded region. a. y 3 2 1-4 -3-2 -1 1 2 3 4 5 x -1 Math 1313 - Prerequisites 4
b. 3 y 2 1-3 -2-1 1 2 3 4 5 6 x -1 c. The slope of the solid line with positive slope is 1. The slope of the dashed line is y 2. 5 3 2 1 3 2 1 1 2 3 4 5 6 x 1 Math 1313 - Prerequisites 5
Example 8: Which of the following are true? 7 3 I. The point, is on the graph of 10 x 2y > 6. 10 16 II. The following two lines are perpendicular to each other. Line L : 2x + 5y = 5 1 Line L 2 : passes through (4, 6) and 6,2. 5 III. A line through (3, 4) and (3, 5) is a horizontal line. IV. The line y = 2x 2 rises to the left. V. The point (1, -3) is a point in the solution set of x > 0 and y 0. Example 9: Which of the following are false? I. The slope of the line through (3, -2) and (5, -2) is undefined. Math 1313 - Prerequisites 6
II. The point 5,0 4 is on the graph of 4 x 7y > 5. III. Line L 1 has slope m. Line L 2 is parallel to line L 1, so L 2 has slope m. IV. The point (1, 2) is in the solution set of y > x 1. V. The solution to 2x 3y 4 is the half-plane lying below the line 2 x 3y = 4. Math 1313 - Prerequisites 7