3 1 lines
DESCRIPTION
TRANSCRIPT
3.1 Lines3.1 Lines
Lines
3.1 Lines3.1 Lines
• Linear equation – a polynomial equation of the first degree
• The term “linear” stems from the fact that the graph of such an equation is a straight line.
• Forms:
€
y = mx + b (Slope- intercept form)
€
Ax + By + C = 0 (General form)
3.1 Lines3.1 Lines
To graph a line, we generate a pair of points using its equation and then connect the two points.
Example. If the linear equation is , a pair of points is generated when we let x = 0 and then x = 1.
€
2x + 3y − 9 = 0
€
1
€
0
€
x
€
y = − 23 x + 3
€
3
€
73
3.1 Lines3.1 Lines
Connecting the two points (0,1) and give us the graph of the line.
€
1, 73( )
3.1 Lines3.1 Lines
Special cases:• y = b
• x = a
€
- a horizontal line passing through(0,b)
€
- a vertical line passing through(a,0)
3.1 Lines3.1 Lines
• The slope of a line describes its incline.• The higher the value of the slope, the steeper
the incline is.• The slope is also defined as a rate of change
(the ratio of the change in y coordinate to the change in x coordinate between any two points on the line).
3.1 Lines3.1 Lines
If the line is not vertical and (x1, y1) and (x2, y2) are distinct points on the line, then the slope of the line is
€
m =y2 − y1
x2 − x1
3.1 Lines3.1 Lines
Example. Find the slope of the line that passes through the points (-1,0) and (3,8).
The slope m is given by
€
m =y2 − y1
x2 − x1
=8 − 0
3 − (−1)=
8
4= 2
3.1 Lines3.1 Lines
• The slope of a horizontal line is zero while that of a vertical line is not defined.
• Two non-vertical lines are parallel if and only if m1 = m2.
• Two lines are perpendicular if and only if m1m2 = -1.
3.1 Lines3.1 Lines
Example. What is the slope of the line parallel to the line whose equation is ?
Rewrite the given equation into the form y = mx + b. The slope is the coefficient m of x.
Hence, the slope of the given line is 2.Since two parallel lines have equal slopes, the
other line must also have a slope of 2.
€
2x − y − 4 = 0€
2x − y − 4 = 0
€
y = 2x − 4
3.1 Lines3.1 Lines
• Linear equations can be rewritten into several different forms. These forms are collectively referred to as “equations of the straight line”.
• Slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept
• Illustration. The equation of the line with slope 2 and y-intercept –5 is
€
or 2x − y − 5 = 0
€
y = 2x − 5
3.1 Lines3.1 Lines
Two-point form: ,
with.the line passing through the points (x1, y1) and (x2, y2)
Illustration. The equation of the line which passes through (2,1) and (-1,5) is
€
y − y1 =y2 − y1
x2 − x1
(x − x1)
€
y −1 =5 −1
−1− 2(x − 2)
€
y −1 = 4−3 (x − 2)
€
−3y + 3 = −4 x + 8
€
4x − 3y − 5 = 0
3.1 Lines3.1 Lines
Point-slope form: y – y1 = m(x – x1), with the line having a slope m and passing through the point (x1, y1)
Illustration. The equation of the line whose slope is and passes through (3,5) is
€
y − 5 = − 12 (x − 3)
€
2y −10 = −x + 3
€
x + 2y −13 = 0€
−12
3.1 Lines3.1 Lines
Intercept form: , with x-intercept a
and. y-intercept b
Illustration. The equation of the line with y-intercept 5 and x-intercept -1 is
€
5x − y = −5
€
5x − y + 5 = 0
€
x
a+
y
b=1
€
x
−1+
y
5=1
3.1 Lines3.1 Lines
1.a Graph the line with slope and passing through the point (1,4).
From (1,4), move 2 units up and then 3 units to the right. Connect the points.€
m = 23
3.1 Lines3.1 Lines
1.b Graph the line with slope and passing through the point (2,5).
From (2,5), move 1 unit up and then 4 units to the left (or 1 unit down and then 4 units to the right)
€
m = − 14
3.1 Lines3.1 Lines
2.a Find the slope and y-intercept of the line 2x – y = 4.
€
2x − y = 4
€
y = 2x − 4
€
⇒ m = 2, b = −4
3.1 Lines3.1 Lines
2.d Find the slope and y-intercept of the line 3(y + 1) = 2(x – 5).
€
3(y +1) = 2(x − 5)
€
3y + 3 = 2x −10
€
⇒ m = 23 , b = 13
3
€
3y = 2x −13
€
y = 23 x − 13
3
3.1 Lines3.1 Lines
3.a Find an equation of the line passing through (4,3) and (2,5). Express your answer in slope-intercept form.
€
⇒ y − 3 =5 − 3
2 − 4(x − 4)
€
y − 3 = 2−2 (x − 4)
€
y − 3 = −(x − 4)€
y − y1 =y2 − y1
x2 − x1
(x − x1)
€
y = −x + 4 + 3
€
y = −x + 7
3.1 Lines3.1 Lines
3.d Find an equation of the line passing through (3,2) and has slope 3. Express your answer in slope-intercept form.
€
⇒ y − 2 = 3(x − 3)
€
y = 3x − 9 + 2
€
y = 3x − 7€
y − y1 = m(x − x1)
3.1 Lines3.1 Lines
3.g Find an equation of the line with slope 4 and y-intercept 2. Express your answer in slope-intercept form.
€
⇒ y = 4x + 2
€
y = mx + b
3.1 Lines3.1 Lines
4.a Graph .
Draw a line through (0,200) and (1,225).
€
y = 25x + 200, x ≥ 0
3.1 Lines3.1 Lines
5.a Find an equation of the line passing through (-2,4) and is perpendicular to the line 4x + 3y = 2. Express your answer in slope-intercept form.
€
4x + 3y = 2
€
3y = −4x + 2
€
y = − 43 x + 2
3
€
m = − 43
€
⇒ m⊥= 34
€
y − 4 = 34 (x − (−2))
€
4(y − 4) = 3(x + 2)
€
4y −16 = 3x + 6
€
y = 34 x + 11
2
3.1 Lines3.1 Lines
5.c Find an equation of the line passing through (1,4) and is parallel to the line -4x + 6y = 2. Express your answer in slope-intercept form.
€
−4x + 6y = 2
€
6y = 4x + 2
€
y = 23 x + 1
3
€
m = 23
€
⇒ m|| = 23
€
y − 4 = 23 (x −1)
€
3(y − 4) = 2(x −1)
€
3y −12 = 2x − 2
€
y = 23 x + 10
3
3.1 Lines3.1 Lines
5.g Find an equation of the line passing through (4,-3) and has a slope of 0. Express your answer in slope-intercept form.
If the slope is 0, the line is horizontal.So our line is a horizontal line passing through
(4,-3).The equation is y = –3.
3.1 Lines3.1 Lines
6.a Are the following pairs of lines parallel, perpendicular, or neither?
€
6x + 3y = 4
2x + y = −5
€
6x + 3y = 4
3y = −6x + 4
y = −2x + 43
€
2x + y = −5
y = −2x − 5
€
The lines are parallel because they have the same slope.