3 5 Slopesofof Lines Warm Up Lesson Presentation

3 -5 Slopesofof. Lines Warm Up Lesson Presentation Lesson Quiz Holt Geometry Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Warm Up Find the value of m. 1. 2. 3. 4. undefined Holt Mc. Dougal Geometry 0

3 -5 Slopes of Lines Objectives Find the slope of a line. Use slopes to identify parallel and perpendicular lines. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Vocabulary rise run slope Holt Mc. Dougal Geometry

3 -5 Slopes of Lines The slope of a line in a coordinate plane is a number that describes the steepness of the line. Any two points on a line can be used to determine the slope. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Example 1 A: Finding the Slope of a Line Use the slope formula to determine the slope of each line. AB Substitute (– 2, 7) for (x 1, y 1) and (3, 7) for (x 2, y 2) in the slope formula and then simplify. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Example 1 B: Finding the Slope of a Line Use the slope formula to determine the slope of each line. AC Substitute (– 2, 7) for (x 1, y 1) and (4, 2) for (x 2, y 2) in the slope formula and then simplify. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Example 1 C: Finding the Slope of a Line Use the slope formula to determine the slope of each line. AD Substitute (– 2, 7) for (x 1, y 1) and (– 2, 1) for (x 2, y 2) in the slope formula and then simplify. The slope is undefined. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Remember! A fraction with zero in the denominator is undefined because it is impossible to divide by zero. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Example 1 D: Finding the Slope of a Line Use the slope formula to determine the slope of each line. CD Substitute (4, 2) for (x 1, y 1) and (– 2, 1) for (x 2, y 2) in the slope formula and then simplify. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Check It Out! Example 1 Use the slope formula to determine the slope of JK through J(3, 1) and K(2, – 1). Substitute (3, 1) for (x 1, y 1) and (2, – 1) for (x 2, y 2) in the slope formula and then simplify. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines One interpretation of slope is a rate of change. If y represents miles traveled and x represents time in hours, the slope gives the rate of change in miles per hour. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Example 2: Transportation Application Justin is driving from home to his college dormitory. At 4: 00 p. m. , he is 260 miles from home. At 7: 00 p. m. , he is 455 miles from home. Graph the line that represents Justin’s distance from home at a given time. Find and interpret the slope of the line. Use the points (4, 260) and (7, 455) to graph the line and find the slope. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Example 2 Continued The slope is 65, which means Justin is traveling at an average of 65 miles per hour. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Check It Out! Example 2 What if…? Use the graph below to estimate how far Tony will have traveled by 6: 30 P. M. if his average speed stays the same. Since Tony is traveling at an average speed of 60 miles per hour, by 6: 30 P. M. Tony would have traveled 390 miles. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines What does it mean when lines have equal slopes? Draw an example. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines How can you tell if lines are perpendicular? Draw an example. Using their slopes? Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Holt Mc. Dougal Geometry

3 -5 Slopes of Lines If a line has a slope of perpendicular line is The ratios Holt Mc. Dougal Geometry and , then the slope of a. are called opposite reciprocals.

3 -5 Slopes of Lines Caution! Four given points do not always determine two lines. Graph the lines to make sure the points are not collinear. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Example 3 A: Determining Whether Lines Are Parallel, Perpendicular, or Neither Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, or neither. UV and XY for U(0, 2), V(– 1, – 1), X(3, 1), and Y(– 3, 3) The products of the slopes is – 1, so the lines are perpendicular. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Example 3 B: Determining Whether Lines Are Parallel, Perpendicular, or Neither Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, or neither. GH and IJ for G(– 3, – 2), H(1, 2), I(– 2, 4), and J(2, – 4) The slopes are not the same, so the lines are not parallel. The product of the slopes is not – 1, so the lines are not perpendicular. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Example 3 C: Determining Whether Lines Are Parallel, Perpendicular, or Neither Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, or neither. CD and EF for C(– 1, – 3), D(1, 1), E(– 1, 1), and F(0, 3) The lines have the same slope, so they are parallel. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Check It Out! Example 3 a Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. WX and YZ for W(3, 1), X(3, – 2), Y(– 2, 3), and Z(4, 3) Vertical and horizontal lines are perpendicular. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Check It Out! Example 3 b Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. KL and MN for K(– 4, 4), L(– 2, – 3), M(3, 1), and N(– 5, – 1) The slopes are not the same, so the lines are not parallel. The product of the slopes is not – 1, so the lines are not perpendicular. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Check It Out! Example 3 c Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. BC and DE for B(1, 1), C(3, 5), D(– 2, – 6), and E(3, 4) The lines have the same slope, so they are parallel. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Lesson Quiz 1. Use the slope formula to determine the slope of the line that passes through M(3, 7) and N(– 3, 1). m=1 Graph each pair of lines. Use slopes to determine whether they are parallel, perpendicular, or neither. 2. AB and XY for A(– 2, 5), B(– 3, 1), X(0, – 2), and Y(1, 2) 4, 4; parallel 3. MN and ST for M(0, – 2), N(4, – 4), S(4, 1), and T(1, – 5) Holt Mc. Dougal Geometry

3 -5 Slopes of Lines The equation of a line can be written in many different forms. The point-slope and slope-intercept forms of a line are equivalent. Because the slope of a vertical line is undefined, these forms cannot be used to write the equation of a vertical line. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Remember! A line with y-intercept b contains the point (0, b). A line with x-intercept a contains the point (a, 0). Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Example 1 A: Writing Equations In Lines Write the equation of each line in the given form. the line with slope 6 through (3, – 4) in pointslope form y – y 1 = m(x – x 1) y – (– 4) = 6(x – 3) Holt Mc. Dougal Geometry Point-slope form Substitute 6 for m, 3 for x 1, and -4 for y 1.

3 -5 Slopes of Lines Example 1 B: Writing Equations In Lines Write the equation of each line in the given form. the line through (– 1, 0) and (1, 2) in slopeintercept form Find the slope. y = mx + b Slope-intercept form 0 = 1(-1) + b Substitute 1 for m, -1 for x, and 0 for y. 1=b y=x+1 Holt Mc. Dougal Geometry Write in slope-intercept form using m = 1 and b = 1.

3 -5 Slopes of Lines Example 1 C: Writing Equations In Lines Write the equation of each line in the given form. the line with the x-intercept 3 and y-intercept – 5 in point slope form Use the point (3, -5) to find the slope. y – y 1 = m(x – x 1) y – 0 = 5 (x – 3) 3 5 y = 3 (x - 3) Holt Mc. Dougal Geometry Point-slope form Substitute 5 for m, 3 for x , and 0 3 for y. 1 Simplif y. 1

3 -5 Slopes of Lines Check It Out! Example 1 a Write the equation of each line in the given form. the line with slope 0 through (4, 6) in slopeintercept form y – y 1 = m(x – x 1) Point-slope form y – 6 = 0(x – 4) Substitute 0 for m, 4 for x 1, and 6 for y 1. y=6 Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Check It Out! Example 1 b Write the equation of each line in the given form. the line through (– 3, 2) and (1, 2) in pointslope form Find the slope. y – y 1 = m(x – x 1) Point-slope form y – 2 = 0(x – 1) Substitute 0 for m, 1 for x 1, and 2 for y 1. y-2=0 Simplif y. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines A system of two linear equations in two variables represents two lines. The lines can be parallel, intersecting, or coinciding. Lines that coincide are the same line, but the equations may be written in different forms. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Example 3 A: Classifying Pairs of Lines Determine whether the lines are parallel, intersect, or coincide. y = 3 x + 7, y = – 3 x – 4 The lines have different slopes, so they intersect. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Example 3 B: Classifying Pairs of Lines Determine whether the lines are parallel, intersect, or coincide. Solve the second equation for y to find the slopeintercept form. 6 y = – 2 x + 12 Both lines have a slope of , and the y-intercepts are different. So the lines are parallel. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Example 3 C: Classifying Pairs of Lines Determine whether the lines are parallel, intersect, or coincide. 2 y – 4 x = 16, y – 10 = 2(x - 1) Solve both equations for y to find the slopeintercept form. 2 y – 4 x = 16 2 y = 4 x + 16 y = 2 x + 8 y – 10 = 2(x – 1) y – 10 = 2 x - 2 y = 2 x + 8 Both lines have a slope of 2 and a y-intercept of 8, so they coincide. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Check It Out! Example 3 Determine whether the lines 3 x + 5 y = 2 and 3 x + 6 = -5 y are parallel, intersect, or coincide. Solve both equations for y to find the slopeintercept form. 3 x + 5 y = 2 3 x + 6 = – 5 y 5 y = – 3 x + 2 Both lines have the same slopes but different y-intercepts, so the lines are parallel. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Example 4: Problem-Solving Application Erica is trying to decide between two car rental plans. For how many miles will the plans cost the same? Holt Mc. Dougal Geometry

3 -5 Slopes of Lines 1 Understand the Problem The answer is the number of miles for which the costs of the two plans would be the same. Plan A costs $100. 00 for the initial fee and $0. 35 per mile. Plan B costs $85. 00 for the initial fee and $0. 50 per mile. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines 2 Make a Plan Write an equation for each plan, and then graph the equations. The solution is the intersection of the two lines. Find the intersection by solving the system of equations. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines 3 Solve Plan A: y = 0. 35 x + 100 Plan B: y = 0. 50 x + 85 0 = – 0. 15 x + 15 Subtract the second equation from the first. x = 100 Solve for x. y = 0. 50(100) + 85 = 135 Substitute 100 for x in the first equation. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines 4 Look Back Check your answer for each plan in the original problem. For 100 miles, Plan A costs $100. 00 + $0. 35(100) = $100 + $35 = $135. 00. Plan B costs $85. 00 + $0. 50(100) = $85 + $50 = $135, so the plans cost the same. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Check It Out! Example 4 What if…? Suppose the rate for Plan B was also $35 per month. What would be true about the lines that represent the cost of each plan? The lines would be parallel. Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Lesson Quiz: Part I Write the equation of each line in the given form. Then graph each line. 1. the line through (-1, 3) and (3, -5) in slopeintercept form. y = – 2 x + 1 2. the line through (5, – 1) with slope in point-slope form. y + 1 = 2 (x – 5) 5 Holt Mc. Dougal Geometry

3 -5 Slopes of Lines Lesson Quiz: Part II Determine whether the lines are parallel, intersect, or coincide. 3. y – 3 = – 1 x, y – 5 = 2(x + 3) 2 intersect 4. 2 y = 4 x + 12, 4 x – 2 y = 8 parallel Holt Mc. Dougal Geometry