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 2. m = –1, x = 5, and y = –4 4. 4x – 2y = 8

Objectives Graph lines and write their equations in slope-intercept and point-slope form. Classify lines as parallel, intersecting, or coinciding.

The equation of a line can be written in many different forms 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.

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 each line in the given form. the line with slope 6 through (3, –4) in point-slope form

Example 1B: Writing Equations In Lines Write the equation of each line in the given form. the line through (–1, 0) and (1, 2) in slope-intercept form

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)

A system of two linear equations in two variables represents two 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.

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

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.

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? 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

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.

Classwork/Homework Pg. 194 (2-10 even, 13-31, 33-44, 47-50)