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4.4 Slope - review + new

slope m=y2−y1x2−x1

=y1− y2x1−x2

=riserun

2

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

3

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.

4

4.4 Slope - review + new

Two lines can be parallel to each other:

then their slopes are equal

m1=m2

5

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