1. 3.4 Notes: Equations of Lines
How can you write an equation of a line that is parallel or perpendicular to a given line and passes through a given point?

2. Vocab! 𝒎= slope and 𝒃= y-intercept 𝒚− 𝒚 𝟏 =𝒎 𝒙− 𝒙 𝟏  
Slope-Intercept Form
 𝒚=𝒎𝒙+𝒃
𝒎= slope and 𝒃= y-intercept
Point-Slope Form
 𝒚− 𝒚 𝟏 =𝒎 𝒙− 𝒙 𝟏
Where 𝒙 𝟏 , 𝒚 𝟏 is a point and 𝒎= the slope

3. Example 2
a) Write an equation in slope-intercept form of the line with slope of 6 and y-intercept of –3. Then graph the line.
𝑚=6
𝑏=−3
𝑦=𝑚𝑥+𝑏
𝑦=6𝑥−3

4. Example 2 cont.
b) Write an equation in slope-intercept form of the line with slope of –1 and y-intercept of 4.
𝑚=−1
𝑏=4
𝑦=𝑚𝑥+𝑏
𝑦=−1𝑥+4

5. Example 3
a) Write an equation in point-slope form of the line whose slope is − 3 5 that contains (–10, 8). Then graph the line.
b) Write an equation in point-slope form of the
line whose slope is that contains (6, –3).
Slope = − 𝟑 𝟓
Point = (−𝟏𝟎, 𝟖)
Point slope form = 𝒚− 𝒚 𝟏 =𝒎(𝒙− 𝒙 𝟏 )
𝒚−𝟖=− 𝟑 𝟓 (𝒙+𝟏𝟎)
Slope = 𝟏 𝟑
Point = (𝟔, −𝟑)
Point slope form = 𝒚− 𝒚 𝟏 =𝒎(𝒙− 𝒙 𝟏 )
𝒚+𝟑= 𝟏 𝟑 (𝒙−𝟔)

6. You Try!
1. Write an equation in slope-intercept form for a line containing (4, 9) and (–2, 0).
Find slope using given points.
0−9 −2−4 = −9 −6 = 3 2
4. Write equation in slope-intercept form where 𝑚= 3 2 and 𝑏=3
𝑦= 3 2 𝑥+3
2. Put slope into slope-intercept form.
𝑦= 3 2 𝑥+𝑏
3. Using one of the points [I used (4, 9)], plug it back in to solve for b
9= 𝑏
𝑏=3

7. You Try!
2. Write an equation in slope-intercept form for a line containing (–3, –7) and (–1, 3). Try it first! Answer is on next slide.

8. You Try!
2. Write an equation in slope-intercept form for a line containing (–3, –7) and (–1, 3).
Find slope using given points.
3−−7 −1−−3 = 10 2 =5
4. Write equation in slope-intercept form where 𝑚=5 and 𝑏=8
𝑦=5𝑥+8
2. Put slope into slope-intercept form.
𝑦=5𝑥+𝑏
3. Using one of the points [I used (-1, 3)], plug it back in to solve for b
3=5(−1)+𝑏
𝑏=8

9. Try it first! Answer is on next slide.
You Try!
3. Write an equation of the line through (5, –2) and (0, –2) in slope-intercept form.
Try it first! Answer is on next slide.

10. You Try!
3. Write an equation of the line through (5, –2) and (0, –2) in slope-intercept form.
Find slope using given points.
−2−−2 0−5 = 0 −5 =0
4. Write equation in slope-intercept form where 𝑚=0 and 𝑏=−2
𝑦=0𝑥−2
𝑦=−2
2. Put slope into slope-intercept form.
𝑦=0𝑥+𝑏
3. Using one of the points [I used (5, -2)], plug it back in to solve for b
−2=0(5)+𝑏
𝑏=−2

11. Horizontal Line
𝑦=𝑏 where 𝑏 is the y-intercept
𝑦=−3

12. Vertical Line
x=𝑎 where 𝑎 is the x-intercept
𝑥=1

13. Steps in writing a line PERPENDICULAR to a given line through a point
Find the perpendicular slope from the original equation
Write the equation in slope-intercept form
Find b using given point
Write new equation with perpendicular slope and b

14. Example 5
a) Write an equation in slope-intercept form for a line perpendicular to the line 𝑦= 1 5 𝑥+2 through (2, 0).
Perpendicular slope = -5
Write in slope-intercept form
𝑦=−5𝑥+𝑏
Plug in point (2, 0) to find b
0=−5 2 +𝑏
𝑏=10
Write equation in slope-intercept form
𝑦=−5𝑥+10

15. Example 5 cont.
b) Write an equation in slope-intercept form for a line perpendicular to the line 𝑦= 1 3 𝑥+2 through (0, 8).
Perpendicular slope = -3
Write in slope-intercept form
𝑦=−3𝑥+𝑏
Plug in point (0, 8) to find b
8=−3 0 +𝑏
𝑏=8
Write equation in slope-intercept form
𝑦=−3𝑥+8

16. You Try!
Write the equation in slope-intercept form of the line parallel and line perpendicular to given line through given point. Parallel Perpendicular 1) y = 4x + 7 (─2, ─9) ____________________ __________________ 2) 2x ─ 5y = 10 (3, ─7) _____________________ _________________
𝑦=− 1 4 𝑥−9.5
𝑦=4𝑥−1
1. Perpendicular = opposite reciprocal slope (− 1 4 )
2. Plug in point to find b
−9=− 1 4 (−2)+𝑏
𝑏=−9.5
1. Parallel = same slope (4)
2. Plug in point to find b
−9=4 −2 +𝑏
𝑏=−1
𝑦= 2 5 𝑥−8.2
𝑦=− 5 2 𝑥+.5
𝑦= 2 5 𝑥−2 (change to slope-intercept)
1. Perpendicular = opposite reciprocal slope (− 5 2 )
2. Plug in point to find b
−7=− 5 2 (3)+𝑏
𝑏=.5
1. Parallel = same slope ( 2 5 )
2. Plug in point to find b
−7= 𝑏
𝑏=−8.2

17. Opposite reciprocal = perpendicular
You Try!
State if the lines are Parallel, Perpendicular, or Neither Change to slope-intercept form to find their slope 3) 6x ─ 12y = 24 4) 4x + y = 5 5) ─2x + 7y = 14 4x + 2y = 8 3x +12y = ─6 4x = 14y
6𝑥−12𝑦=24
𝑦= 1 2 𝑥−2
4𝑥+𝑦=5
𝑦=−4𝑥+5
−2𝑥+7𝑦=14
𝑦= 2 7 𝑥+2
3𝑥+12𝑦=−6
𝑦=− 1 4 𝑥− 1 2
4𝑥=14𝑦
𝑦= 2 7 𝑥
4𝑥+2𝑦=8
𝑦=−2𝑥−4
Opposite reciprocal = perpendicular
Neither
Parallel