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Slopes
The slope m of the line passing through the points (x
(^1) ,y (^1) ) and
(x (^2) ,y (^2) ) is given by m =
2 y^ (^) – y 1
2 x^ (^) – x (^1).
a) (–1, 2) and (3, 5)Example: Find the slope of the line passing though the given points: m =
b) (0, 1) and (2, 6) m =
c) (5, 1) and (1, 3) m =
d) (2, –3) and (–1, –9) m =
e) (1, 3) and (4, 6) m =
f) (3, 6) and (1, 6) m =
g) (–3, 2) and (–3, 5) m =
slope undefined
Examples:
Find the slope and the y–intercept of each of the following
a) –2x + 3y = 6lines: y =3y = 2x + 6 3 2 (^) x + 2
y–intercept 2slope 2/
b) y = –3x + 2 y–intercept 2slope –
c) 5x – 4y = 20 y =–4y = –5x + 20 4 5 (^) x – 5
y–intercept –5slope 5/
d) 3y + 6 = 0 y–intercept –2slope 0y = –23y = –
e) y = 6 y–intercept 6slope 0
f) x = 3 no y–interceptslope undefined
Finding Equations Of Lines
To find the equation of any line, you always need two types of
information: slope information and point information.
Point–Slope
y – y
1 = m(x – x
Slope–Intercept
y = mx + b
Horizontal line
y = b
Vertical line
x = a
a. The slope of the line passing through two points is m =
2 y^ (^) – y 1
x (^2)
c. Slopes of perpendicular lines are negative reciprocals.b. Slopes of parallel lines are equal.
Examples:
Find the equation of the line: (Put answers in slope–intercept
a) through (2, 3) with slope –1/2form if possible) y – (3) = –
2 1 (^) (x – 2)
y – 3 = –
2 1 (^) x + 1
y = –
2 1 (^) x + 4
b) through (–4, –1) with slope 2 y + 1 = 2x + 8y + 1 = 2(x + 4)y – (–1) = 2(x – (–4)) y = 2x + 7
c) through (–1, 2) and (4, –2) m
5 4
y – 2 = –
5 4 (^) (x – (–1))
y – 2 = –
5 4 (^) (x + 1)
y – 2 = –
5 4 (^) x – 5 4
y = –
5 4 x + 5 6
d) through (6, –2) and (2, 0) m
2 1
y – 0 = –
2 1 (x – 2)
y = –
2 1 (^) x + 1
e) through (2, 3) and horizontal
y = 3
f) through (–2, 6) and vertical
x = –
c) x + y > 3
d) y > 1
(^) x
y
(^) x
y
e) x + 3y
f) x (^) ≤ (^0)
(^) x
y
(^) x
y
