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4.4 Slope - review + new
slope m=y2−y1x2−x1
=y1− y2x1−x2
=riserun
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4.4 Slope - review + new
Forms of linear equations in two variables:
Ax+Bx = C standard formy = mx + b slope-intercept form
slope y-intercept
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4.4 Slope - review + new
m1=−1m2
m1⋅m2=−1
or
Two lines can be perpendicular to each other:
then their slopesare negative reciprocals, i.e.
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4.4 Slope - review + new
Two lines can be parallel to each other:
then their slopes are equal
m1=m2
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4.4 Slope - review + new
Example: let’s check whether the given by equations or by points lines are parallel, perpendicular or neither.
a) 4x+5y = 8 and 10x-8y = 3
b) one line passes through the points (1,2) and (3,-1),another line passes through the points (0,1) and (-2,4)
c) 3x+2y = 5 and 4x+5y = 8
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4.4 Slope - review + new
Example: let’s check whether the given by equations or by points lines are parallel, perpendicular or neither.
a) 4x+5y = 8 and 10x-8y = 3-4x +8y5y = 8 – 4x 10x = 3 + 8y5 5 -3 -3
10x – 3 = 8y8 8y=
8−4 x5
y=−4 x5
+85
10 x−38
= y y=10 x8
−38
m1=−45
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4.4 Slope - review + new
Example: let’s check whether the given by equations or by points lines are parallel, perpendicular or neither.
a) 4x+5y = 8 and 10x-8y = 3
y=−4 x5
+85
y=10 x8
−38
m1=−45
m2=108
=54
m1⋅m2=−45
×54=−1 The lines are perpendicular
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4.4 Slope - review + new
Example: let’s check whether the given by equations or by points lines are parallel, perpendicular or neither.
b) one line passes through the points (1,2) and (3,-1),another line passes through the points (0,1) and (-2,4)
m1=−1−23−1
=−32
m2=4−1
−2−0=−
32
m1=m2 hence the lines are parallel
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4.4 Slope - review + new
Example: let’s check whether the given by equations or by points lines are parallel, perpendicular or neither.
c) 3x+2y = 5 and 4x+5y = 8 -3x -4x
2y = 5 – 3x 5y = 8 – 4x 2 2 5 5
y=52−3 x2
y=85−4 x5
neither
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4.5 The point-slope form of the equation
Knowing a point (x0,y0) and a slope we can use the point-slope equation of a line y-y
0 = m(x-x
0)
(x0,y0)
(x,y)
x
y
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4.5 The point-slope form of the equation
Knowing a point (x0,y0) and a slope we can use the point-slope equation of a line y-y
0 = m(x-x
0)
Derivation:
(x0,y0)
(x,y)
x
y
m=y−y0x−x0
*(x-x0)(x-x
0)*
(x−x0)m= y− y0y− y0=(x−x0)m
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Summary
Forms of linear equations in two variables:
Ax+Bx = C standard formy = mx + b slope-intercept form
y-y0 = m(x-x
0) slope-intercept form
slope y-intercept
If two lines are perpendicular, then their slopes are negative reciprocals, i.e.
If two lines are parallel, then their slopes are equal, i.e.
m1⋅m2=−1 m1=−1m2
m1=m2