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?

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

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

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

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 1 3 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)

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= 3 2 4 +𝑏 𝑏=3

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.

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

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.

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

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

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

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

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

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

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= 2 5 3 +𝑏 𝑏=−8.2

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