3 6 Linesininthe the Coordinate Plane Warm Up

3 -6 Linesininthe the. Coordinate. Plane Warm Up Lesson Presentation Lesson Quiz Holt Geometry Holt Mc. Dougal Geometry

3 -6 Lines in the Coordinate Plane Warm Up Substitute the given values of m, x, and y into the equation y = mx + b and solve for b. 1. m = 2, x = 3, and y = 0 b = – 6 2. m = – 1, x = 5, and y = – 4 b = 1 Solve each equation for y. 3. y – 6 x = 9 y = 6 x + 9 4. 4 x – 2 y = 8 y = 2 x – 4 Holt Mc. Dougal Geometry

3 -6 Lines in the Coordinate Plane Objectives Graph lines and write their equations in slope -intercept and point-slope form. Classify lines as parallel, intersecting, or coinciding. Holt Mc. Dougal Geometry

3 -6 Lines in the Coordinate Plane Vocabulary point-slope form slope-intercept form Holt Mc. Dougal Geometry

3 -6 Lines in the Coordinate Plane 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 -6 Lines in the Coordinate Plane Holt Mc. Dougal Geometry

3 -6 Lines in the Coordinate Plane 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 -6 Lines in the Coordinate Plane 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 -6 Lines in the Coordinate Plane 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 -6 Lines in the Coordinate Plane 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) 5 y– 0= (x – 3) 3 5 y = 3 (x - 3) Holt Mc. Dougal Geometry Point-slope form 5 Substitute 3 0 for y 1. Simplify. for m, 3 for x 1, and

3 -6 Lines in the Coordinate Plane 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 -6 Lines in the Coordinate Plane 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 Simplify. Holt Mc. Dougal Geometry

3 -6 Lines in the Coordinate Plane Example 2 A: Graphing Lines Graph each line. The equation is given in the slope-intercept form, with a slope of and a y-intercept of 1. Plot the point (0, 1) and then rise 1 and run 2 to find another point. Draw the line containing the points. Holt Mc. Dougal Geometry run 2 rise 1 (0, 1)

3 -6 Lines in the Coordinate Plane Example 2 B: Graphing Lines Graph each line. y – 3 = – 2(x + 4) The equation is given in the point-slope form, with a slope of through the point (– 4, 3). Plot the point (– 4, 3) and then rise – 2 and run 1 to find another point. Draw the line containing the points. Holt Mc. Dougal Geometry rise – 2 (– 4, 3) run 1

3 -6 Lines in the Coordinate Plane Example 2 C: Graphing Lines Graph each line. y = – 3 The equation is given in the form of a horizontal line with a y-intercept of – 3. The equation tells you that the y-coordinate of every point on the line is – 3. Draw the horizontal line through (0, – 3). Holt Mc. Dougal Geometry (0, – 3)

3 -6 Lines in the Coordinate Plane Check It Out! Example 2 a Graph each line. y = 2 x – 3 The equation is given in the slope-intercept form, with a slope of and a y-intercept of – 3. Plot the point (0, – 3) and then rise 2 and run 1 to find another point. Draw the line containing the points. Holt Mc. Dougal Geometry run 1 rise 2 (0, – 3)

3 -6 Lines in the Coordinate Plane Check It Out! Example 2 b Graph each line. The equation is given in the point-slope form, with a slope of through the point (– 2, 1). Plot the point (– 2, 1)and then rise – 2 and run 3 to find another point. Draw the line containing the points. Holt Mc. Dougal Geometry rise – 2 (– 2, 1) run 3

3 -6 Lines in the Coordinate Plane Check It Out! Example 2 c Graph each line. y = – 4 The equation is given in the form of a horizontal line with a y-intercept of – 4. The equation tells you that the y-coordinate of every point on the line is – 4. Draw the horizontal line through (0, – 4). Holt Mc. Dougal Geometry (0, – 4)

3 -6 Lines in the Coordinate Plane 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 -6 Lines in the Coordinate Plane Holt Mc. Dougal Geometry

3 -6 Lines in the Coordinate Plane 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 -6 Lines in the Coordinate Plane 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 -6 Lines in the Coordinate Plane 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 -6 Lines in the Coordinate Plane 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 -6 Lines in the Coordinate Plane 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 -6 Lines in the Coordinate Plane 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 -6 Lines in the Coordinate Plane 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 -6 Lines in the Coordinate Plane 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 -6 Lines in the Coordinate Plane 3 Solve Continued The lines cross at (100, 135). Both plans cost $135 for 100 miles. Holt Mc. Dougal Geometry

3 -6 Lines in the Coordinate Plane 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 -6 Lines in the Coordinate Plane 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 -6 Lines in the Coordinate Plane 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 -6 Lines in the Coordinate Plane 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