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CP Math
Section 2-5
Point-Slope Form equation of a line.
Quiz 2-3
1. Graph the line y = 2x + 1
2. Graph the line 3x + 4y = 12
3. Graph the line y = 2
4. Graph the line x = 3
2-5 Write Equations of
Lines
What we will learn:
Finding the equation of a line given the following:
1. The slope and the y-intercept
2. A point on the line and the slope of the line.
3.
2 points on the line.
4. A point on the line and a parallel line.
5. A point on the line and a perpendicular line.
Your turn:
1.
What is the equation of this line?
Your turn:
2.
What is the equation of this line?
Your turn:
Name the correct form of the linear equation.
3.
y  2x  2
4.
 2x  y  2
Slope intercept form
Standard form
y = mx + b
Ax + By = C
Two equations for the same line.
How can you tell them apart?
y  2x  2
 2x  y  2
‘y’ is “alone”.
The constant is “alone”.
Two equations for the same line.
Re-write one form as another form
y  2x  2
-2x
-2x
 2x  y  2
Slope-intercept  standard form
Get the constant alone on one side.
Your turn.
5. Convert from: Slope-intercept to standard form
2
y  x 1
3
6. Convert from: standard form to slope-intercept form
 2 x  y  1
Writing Equations of Lines
What two things do you need in order to write the equation
of a line in slope intercept form?
y = mx + b
Slope
y-intercept
The slope of a line is -3/4. The line crosses the y-axis at
-5. What is the equation of the line?
y = mx + b
3
y
x5
4
Your turn.
7. The slope of a line is 2/3. It crosses the y-axis at -1.
What is the equation of this line in slope-intercept form?
2
y  x 1
3
8. The slope of a line is -3. It crosses the y-axis at 7.
What is the equation of this line in slope-intercept form?
y  3x  7
What information is needed?
y = mx + b
Slope and y-intercept
Slope
the slope equation.
y  y1
m
x  x1
Often times it is
written like this:
What property did
we use to go from
here…
…to…
y  y1  m( x  x1 )
…here?
Point Slope Form of a Linear Equation
y  y1  m( x  x1 )
Where
( x1 , y1 ) is a point on the line and
‘m’ is the slope of the line.
Point Slope Form of a Linear Equation
y  y1  m( x  x1 )
Used to find the equation of a line if:
(1) Given a slope and a point on the line.
(write on board)
Finding the equation of a line:
What is the equation of a line that has
a slope of -2 and passes through the point (3, 4) ?
y  y1  m( x  x1 )
y – 4 = -2(x – 3)
convert to slope intercept form
y – 4 = -2x + 6
y = -2x + 10
Finding the equation of a line:
What is the equation of a line that has
a slope of 3 and passes through the point (-1, -2)
?
y  y1  m( x  x1 )
y – (-2) = 3(x – (-1))
convert to slope intercept form
y + 2 = 3(x + 1)
y + 2 = 3x + 3
y = 3x + 1
What information is needed?
y  mx  b Slope and y-intercept
y  y1  m( x  x1 )
Slope and a point
Your turn:
y  y1  m( x  x1 )
9. What is the slope intercept equation of a line that passes
through the point (-4, 2) and has a slope of -4 ?
10. What is the slope intercept equation of a line that passes
through the point (3, -1) and has a slope of 6 ?
Point Slope Form of a Linear Equation
y  y1  m( x  x1 )
Used to find the equation of a line if:
(1) Given a point on the line and the slope of the line.
(write on board)
Finding the equation of a line:
What is the equation of a line that passes through
the points (-3, 4) and (-2, 6) ?
y  y1  m( x  x1 )
We need a point and the slope. Find the slope!
y2  y1
64

m
 2  (3)
x2  x1
2

1
2
Plug slope and one of the points into the equation.
y – 4 = 2(x – (-2))
y – 4 = 2(x + 2)
y – 4 = 2x + 4
y = 2x + 8
Point Slope Form of a Linear Equation
y  y1  m( x  x1 )
Used to find the equation of a line if:
(1) Given a slope and a point on the line.
(2) Given 2 points on the line.
(write on board)
Your turn:
If you were given the following information, what form of
equation (slope-int. or point-slope) would you use to
come up with the specific equation of the line?
11.
Slope = 3, crosses y-axis at (0, -2)
y  mx  b
12.
(slope-intercept form)
Slope = -2, passes thru the point (1, 8)
y  y1  m( x  x1 )
(point – slope form)
Your turn:
13. What is the equation of a line that passes through
points (-1, 5) and (1, -5) ?
14. What is the equation of a line that passes through
points (0, 7) and (3, 5) ?
Parallel and Perpendicular Lines
Parallel lines have the same _______.
Perpendicular lines have ______ that are negative
___________ of each other.
Your turn:
15. What is the slope of the line?
y  2x  3
16. Are the following lines parallel, perpendicular or
neither?
y  2x  3
y  2 x  5
17. Are the following lines parallel, perpendicular
or neither?
2
3
y
3
x4
y   x6
2
18. Are the following lines parallel, perpendicular
or neither?
5
4
3
1
y  x
5
5
y
3
x
5
Equations of Parallel Lines
How do the slopes of parallel lines compare?
Parallel  same slope
Find the equation of a line that is parallel to the line
y = 2x + 1 and passes through the point (4, -1)
What equation form will you use?
y  y1  m( x  x1 )
What is the slope of the other line?
m2
What is the slope of the our line?
m2
Plug the slope and a point into the equation.
y  (1)  2( x  4)
Equations of Parallel Lines
Parallel  same slope
Find the equation of a line that is parallel to the line
y = 2x + 1 and passes through the point (4, -1)
y  y1  m( x  x1 )
y  (1)  2( x  4)
y 1  2x  8
y  2x  9
Convert to slope intercept form.
Point Slope Form of a Linear Equation
y  y1  m( x  x1 )
Used to find the equation of a line if:
(1) Given a slope and a point on the line.
(2) Given 2 points on the line.
(3) Given a parallel line and a point.
(write on board)
Find the equation of the line
Find the equation of a line that is perpendicular to the
line y = 2x – 6 and passes through the point (0, 1)
What equation form will you use?
What is the slope of the other line?
What is the slope of the our line?
y  y1  m( x  x1 )
m2
m1
Plug the slope and a point into the equation.
y  1   1 ( x  0)
2
2
Find the equation of the line
Find the slope intercept form of a line that is perpendicular to
the line: y = 2x – 6 and passes through the point (0, 1)
y  y1  m( x  x1 )
y  1   1 ( x  0)
2
y 1   1 x
2
Convert to slope intercept form.
y   1 x 1
2
Point Slope Form of a Linear Equation
y  y1  m( x  x1 )
Used to find the equation of a line if:
(1) Given a slope and a point on the line.
(2) Given 2 points on the line.
(3) Given a parallel line and a point.
(4) Given a perpendicular line and a point.
(write on board)
Your Turn:
19. Find the slope intercept form of a line
that is perpendicular to the line:
y = -1/3 x – 2
and passes through the point (4, 5)
Three equations for the same line.
Re-write one form as another form
y  4  2( x  1) Point slope  standard form
Get the “x’s and “y’s” alone on one side.
y  4  2 x  2 (distributive property)
-2x
-2x
 2 x  y  4  2
+4
+4
 2x  y  2
Your turn:
20. Convert from Point slope to standard form
y  2  2 ( x  3)
3
21. Given the “point-slope” form of a linear equation,
rewrite the equation in “slope intercept” form.
y  2  3( x  1)
Fractions!!!
1
2
5
1 2
 
5 1
1 10
 
5 5
9

5
1 2 5
  *
5 1 5
Your turn:
22. 2  4
3
23.
3
2
7
Simplify the following:
Point Slope Form of a Linear Equation
Often times fractions are introduced into these equations.
What is the equation of a line that passes through points
(2, -3) and (-1, 7) ?
y  y1  m( x  x1 )
slope.
7  (3)
10
10
m


1  2
3
3
10
y  7   ( x  (1))
3
10
y  7   ( x  1)
3
Point Slope Form of a Linear Equation
Often times fractions are introduced into these equations.
What is the equation of a line that passes through points
(2, -3) and (-1, 7) ?
Step 3: Plug a point and the slope into the equation.
10
y  7   ( x  1)
3
10
10
y7   x
3
3
10
10
y   x 7
3
3
10
10 7
y  x 
3
3 1
10
10 7 3
y  x  *
3
3 1 3
10
10 21
y  x 
3
3 3
10
11
y  x
3
3
Your turn:
24. What is the equation of a line that passes
through the points (-2, 3) and (3, -5) ?
Equations of Parallel Lines
Parallel  same slope
parallel to the line y = 2x + 1 and
passes through the point (4, -1)
Slope = 2
( x1, y1 )  (4,1)
y  y1  m( x  x1 )
y  (1)  2( x  4)
y 1  2x  8
y  2x  9
Your turn:
25. What is the equation of a line that passes
through the point (2, -7) and is parallel to the
line: y = -x - 3