1.4: equations of lines

M(G&M)–10–9 Solves problems on and off the coordinate plane involving distance, midpoint, perpendicular and parallel lines, or slope.

GSE

G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel orperpendicular to a given line that passes through a given point).

CCSS:

Various Forms of an Equation of a Line.

Slope-Intercept Form

Point-Slope Form

slope of the line

intercept

y mx b

m

b y

1 1

1 1

slope of the line

, is any point

y y m x x

m

x y

Write an equation given the slope and y-interceptEXAMPLE 1

Write an equation of the line shown.

SOLUTION

Write an equation given the slope and y-interceptEXAMPLE 1

From the graph, you can see that the slope is m = and the y-intercept is b = –2. Use slope-intercept form to write an equation of the line.

34

y = mx + b Use slope-intercept form.

y = x + (–2)34

Substitute for m and –2 for b.34

y = x (–2)34

Simplify.

GUIDED PRACTICE

Write an equation of the line that has the given slope and y-intercept.

1. m = 3, b = 1

y = x + 13

ANSWER

2. m = –2 , b = –4

y = –2x – 4

ANSWER

3. m = – , b =

34

72

y = – x +34

72

ANSWER

Write an equation given the slope and a pointEXAMPLE 2

Write an equation of the line that passes through (5, 4) and has a slope of –3.

Because you know the slope and a point on the line, use point-slope form to write an equation of the line. Let (x1, y1) = (5, 4) and m = –3.

y – y1 = m(x – x1) Use point-slope form.

y – 4 = –3(x – 5) Substitute for m, x1, and y1.

y – 4 = –3x + 15 Distributive property

SOLUTION

y = –3x + 19 Write in slope-intercept form.

Slope Formula

The slope of a line through the points (x1, y1) and (x2, y2) is as follows:

yy22 –– yy11 xx22 –– xx11

m =

Example 1

• Write the equation of the line with slope = -2 and passing through the point (3, -5).

• Substitute m and into the Point-Slope Formula.

1 1( , )x y

1 1

5 2

( ) ( )

( ) ( )

5 2 6

3

2 1

y y m x x

y x

y x

y x

Example 2: Point Slope Form• Let’s find the equation for the line passing through

the points (3,-2) and (6,10)

Y

X

Y-axis

X-axis

YY

XX

First, Calculate First, Calculate mm : :

m =Y

X=

(10 – -2)

(6 – 3)

(3,-2)

(6,10)

33

1212== == 44

Example 2: Point Slope Form• To find the equation for the line passing through

the points (3,-2) and (6,10)

Y

X

Y-axis

X-axis

YY

XX

y – yy – y11 = = mm(x – x(x – x11))

Next plug it into Point Slope From :Next plug it into Point Slope From :

(3,-2)

(6,10)

y – y – -2-2 = = 44(x – (x – 33))

Select one point as PSelect one point as P11 : :Let’s use Let’s use (3,-2)(3,-2)

The Equation becomes:The Equation becomes:

Example 2: Point Slope Form• If you want ….simplify the equation to put it into Slope Intercept Form

Y

X

Y-axis

X-axis

YY

XX

y + 2 = y + 2 = 44x – 12x – 12Distribute on the right side and the equation becomes:Distribute on the right side and the equation becomes:

(3,-2)

(6,10)

Subtract 2 from both sides gives.Subtract 2 from both sides gives.

y + 2 = y + 2 = 44x – 12x – 12-2 = - 2-2 = - 2

y = 4x – 14y = 4x – 14

Write an equation given two points

EXAMPLE 4

Write an equation of the line that passes through (5, –2) and (2, 10).

SOLUTION

The line passes through (x1, y1) = (5,–2) and (x2, y2) = (2, 10). Find its slope.

y2 – y1m =x2 – x1

10 – (–2) =

2 – 5

12 –3= = –4

Ex3

Write an equation given two points

EXAMPLE 4

You know the slope and a point on the line, so use point-slope form with either given point to write an equation of the line. Choose (x1, y1) = (4, – 7).

y2 – y1 = m(x – x1) Use point-slope form.

y – 10 = – 4(x – 2) Substitute for m, x1, and y1.

y – 10 = – 4x + 8 Distributive property

Write in slope-intercept form.y = – 4x + 8

Example 3

• Write the equation of the line that goes through the points (3, 2) and (5, 4).

1

2 1

4 2

5 32

12

2=y y

mx x

m

m

1 1( ) ( )

( 2) 1( 3)

2 3

1

y y m x x

y x

y x

y x

Understanding Slope

• Two (non-vertical) lines are parallel if and only if they have the same slope. (All vertical lines are parallel.)

• Parallel lines have

the __________ slope

4

2

-2

-4

-6

5

D: (4, -1)

C: (-2, -4)

B: (3, 3)

A: (-1, 1)

Understanding Slope

• The slope of AB is:

• The slope of CD is:

• Since m1=m2, AB || CD

4

2

-2

-4

-6

5

D: (4, -1)

C: (-2, -4)

B: (3, 3)

A: (-1, 1) 1

3 1 2 1

3 1 4 2m

2

1 4 3 1

4 2 6 2m

Perpendicular Lines

• (┴)Perpendicular Lines- 2 lines that intersect forming 4 right angles

Right angle

Slopes of Lines

• 2 lines are their slopes are

opposite reciprocals of each other; such as ½ and -2.

• Vertical and horizontal lines are to each other.

Example• Line l passes through (0,3) and (3,1).

• Line m passes through (0,3) and (-4,-3).

Are they ?

Slope of line l =

Slope of line m =

l m

30

13

3

2-or

3

2

40

33

2

3or

4

6

Opposite Opposite Reciprocals!Reciprocals!

11 xxmyy

Let's look at a line and a point not on the line

(2, 4)

Let's find the equation of a line parallel to y = - x that passes through the point (2, 4)y = - x

What is the slope of the first line, y = - x ?

This is in slope intercept form so y = mx + b which means the slope is –1.

1

-14

Distribute and then solve for y to leave in slope-intercept form.

6 xy

So we know the slope is –1 and it passes through (2, 4). Having the point and the slope, we can use the point-slope formula to find the equation of the line

2

What if we wanted perpendicular instead of parallel?

(2, 4)

Let's find the equation of a line perpendicular to y = - x that passes through the point (2, 4)

y = - xThe slope of the first line is still –1.

11 xxmyy 1 24Distribute and then solve for y to leave in slope-intercept form.

2xy

The slope of a line perpendicular is the negative reciporical so take –1 and "flip" it over and make it negative.

1

1

1

1

So the slope of a perpendicular line is 1 and it passes through (2, 4).

Slope-Intercept Form (y = mx+b)

• Find the equation of a line passing through the points P(0, 2) and Q(3, –2).

2

-2

Q: (3, -2)

P: (0, 2)

•Is this line parallel to a line with the equation

43?

3y x

a) Find the equation of a line that passes through the point G ( -4, 5) and is perpendicular to

b) Write the equation of a line that passes through point P (1, -2) and is parallel the line that passes through points A (-4,6) and B ( 4,10)

53

2

xy

What is the equation of a line that models a line that is

perpendicular to and goes through the point (6,6) ?

43

2 xy

Graphing Equations Conclusions

• What are the similarities you see in the equations for Parallel lines?

• What are the similarities you see in the equations for Perpendicular

lines?

• What are the differences between parallel and perpendicular lines?

• What similarities are there in equations of lines written in slope

intercept and point slope form?

• What difference are there in equations of lines written in slope

intercept and point slope form?

Equation Summary• Slope:

Slope (Slope (mm) =) = Vertical change (YY))

Horizontal change (XX))

Slope-Intercept Form:Slope-Intercept Form:y = y = mmx + x + bb

Point-Slope Form:Point-Slope Form:

y – y – yy11 = = mm(x – (x – xx11))