1. Lines in the Coordinate Plane
3-6
Lines in the Coordinate Plane
Newton
Holt Geometry

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

4. Vocabulary
point-slope form
slope-intercept form
Euclid

5. 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.
Euler

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

8. 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
y – y1 = m(x – x1)
Point-slope form
y – (–4) = 6(x – 3)
Substitute 6 for m, 3 for x1, and -4 for y1.

9. 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
Henri Poincarre

10. Example 1C: 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 – y1 = m(x – x1)
Point-slope form
Substitute for m, 3 for x1, and 0 for y1. 5 3
y – 0 = (x – 3) 5 3
y = (x - 3) 5 3
Simplify.

11. Check It Out! Example 1a
Write the equation of each line in the given form.
the line with slope 0 through (4, 6) in slope-intercept form
y – y1 = m(x – x1)
Point-slope form
Substitute 0 for m, 4 for x1, and 6 for y1.
y – 6 = 0(x – 4)
y = 6

12. Example 2A: Graphing Lines
Graph each line.
Riemann

13. Example 2B: Graphing Lines
Graph each line.
y – 3 = –2(x + 4)
Galois

14. (–4, 3)
rise –2
run 1

15. 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.
Descartes

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

18. Example 3B: Classifying Pairs of Lines
Determine whether the lines are parallel, intersect, or coincide.
Ramanujan

19. Check It Out! Example 3
Determine whether the lines 3x + 5y = 2 and 3x + 6 = -5y are parallel, intersect, or coincide.
Solve both equations for y to find the slope-intercept form.
3x + 5y = 2
3x + 6 = –5y
5y = –3x + 2
Both lines have the same slopes but different y-intercepts, so the lines are parallel.

20. 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 slope-intercept form.
y = –2x + 1
2. the line through (5, –1) with slope in point-slope form. 2 5
y + 1 = (x – 5)

21. Lesson Quiz: Part II
Determine whether the lines are parallel, intersect, or coincide. 1 2
3. y – 3 = – x,
y – 5 = 2(x + 3)
intersect
4. 2y = 4x + 12, 4x – 2y = 8
parallel