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 Solve each equation for y. 3. y – 6x = 9 b = –6 2. m = –1, x = 5, and y = –4 b = 1 y = 6x + 9 4. 4x – 2y = 8 y = 2x – 4

§ 3.6, Lines in the Coordinate Plane 3-6 § 3.6, Lines in the Coordinate Plane Holt McDougal Geometry Holt Geometry

Learning Targets I will graph lines and write their equations in slope-intercept and point-slope form. I will classify lines as parallel, intersecting, or coinciding.

Vocabulary point-slope form slope-intercept form

point-slope form: slope-intercept form:

A line with y-intercept b contains the point (0, b). A line with x-intercept a contains the point (a, 0). Remember!

Example 1A: Writing Equations In Lines Write the equation of the line in slope-intercept form that passes through the points (-1, 0) and (1, 2). Solution: y = mx + b 0 = 1(-1) + b 1 = b y = x + 1

Example 1B: Writing Equations In Lines Write the equation of the line in point-slope form that has the x-intercept 3 and y-intercept -5. y – y1 = m(x – x1) y – 0 = (x – 3) 5 3 y = (x - 3) 5 3

Example 2A: 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. (0, 1) rise 1 run 2

Example 2B: 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. (–4, 3) rise –2 run 1

Example 2C: 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). (0, –3)

Example 2D: Graphing Lines 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). (0, –4)

A system of two linear equations in two variables represents two lines A system of two linear equations in two variables represents two lines. Three conditions exist when graphing two lines: parallel intersecting coinciding (same line, but equations may be written in different forms)

The equations of lines that coincide can be simplified to be exactly the same in the same format. Many times, they are written in different forms.

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

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

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

Critical Thinking: Erica is trying to decide between two car rental plans. For how many miles will the plans cost the same?

Understand the Problem 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.

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.

Solve 3 Plan A: y = 0.35x + 100 Plan B: y = 0.50x + 85 Subtract the second equation from the first. 0 = –0.15x + 15 x = 100 Solve for x. Substitute 100 for x in the first equation. y = 0.50(100) + 85 = 135

Solve Continued 3 The lines cross at (100, 135). Both plans cost $135 for 100 miles.

Look Back 4 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.

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.

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