chapter 7 section 4 slope-intercept and point-slope forms of a linear equation
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Chapter 7 Section 4
Slope-Intercept and Point-Slope Forms of a Linear Equation
Learning Objective• Write a linear equation in slope-intercept form
• Graph a linear equation using the slope and y-intercept
• Use the slope-intercept form to determine the equation of a line.
• Use the point-slope form to determine the equation of a line
• Compare the three methods of graphing linear equation.
• Key Vocabulary: slope, y-intercept, slope-intercept form, point-slope form
Slope-Intercept Form
• Standard form of a linear equation in two variables ax + by = c
• Slope-intercept form of a linear equation y = mx + b where m is the slope, and (0, b) is the y-intercept of the line
• To write an equation in slope-intercept form we solve for y.
• Graph is always a straight line with a slope of m and y-intercept of (0, b)
Slope-Intercept Form Examples
y = 3x – 4 Slope = 3, y-intercept (0, -4) Positive slope , rises from left to
right
y = -2x + 5Slope = -2, y-intercept (0, 5) Negative slope, falls from left to
right
Slope = , y-intercept (0, ) Positive slope, rises from
left to right
1 3
2 2y x
1
2
3
2
Slope-Intercept FormExample: Write the equation 2x + 4y = 8
in slope-intercept form. State the slope and y-intercept.2 4 8
4 -2 8
-2 8 2 8 -
4 4 4
1 1- 2 - - (0, 2)
2 2
x y
y x
xy same as x
y x slope m y intercept
a b a b
c c c
Negative slope Falls from left to rightDown 1 and to the right 2
Perpendicular or Parallel Lines
• Two non-vertical lines with the same slope and different y-intercepts are parallel lines.
• Two lines whose slopes are negative reciprocals of each other are perpendicular lines.
Perpendicular or Parallel LinesExample: Determine if the two lines are perpendicular, parallel, or neither. 3x + y = 5 2y = -6x + 9
Two lines are parallel when their slopes are the same and the y-intercepts are different. When they do not intersect.
1
2
3 5 2 -6 9
-6 9-3 5
29
-32
3
3
x y y x
xy x y
y x
m
m
Parallel
Perpendicular or Parallel LinesExample: Determine if the two lines are perpendicular, parallel, or neither.5x - 4y = 8 4x + 5y = 10
To determine if two lines are perpendicular multiply the slopes of the two lines together. If the product is -1 then the slopes are negative reciprocals, and the lines are perpendicular.
1 2
4 5 8 5 4 10
5 8 4 10
4 55 4
2 24 5
5 4 201
4 5 20
y x y x
x xy y
y x y x
m m Perpendicular
Graph a Linear Equation using Slope and y-intercept
• Method 1: Graph by plotting
• Method 2: Graph by x- and y-intercept x-intercept (x, 0)y-intercept (0, y)
• Method 3: Graph by Slope and y-intercept Form1. Solve the equation for y2. Determine the y-intercept point (0,b)3. If a positive slope we move up and to the right4. If a negative slope we move down and to the right
Graph a Linear Equation using Slope and y-intercept
• Write 2x + 3y = 6 in slope-intercept form; then graph.
2 3 6
3 -2 6
-2 6
3
2- 2
3
2- (0, 2)
3
x y
y x
xy
y x
m
(0,0)
(-3,4)
(0,2)
(3,0)
Down and to the right if negative
Graph a Linear Equation using Slope and y-intercept
• Graph -2x + 5y = 10 using the slope and y-intercept.-2 5 10
5 2 10
2 10
5
22
5
2 (0, 2)
5
x y
y x
xy
y x
m
(0,0)
(0,2)
(5, 4)
Up and to the right if positive
Use the Slope-Intercept FormDetermine the Equation of Line 1
5 1 62
0 3 3
ym
x
(0,0)
(0,-5)
(3, 1)
First: Take two points and use the slope formula to determine the slope
Second: determine the y-intercept (0, b) from the line?
y-intercept (0,-5)
Third: write the formula y = mx + b b is where the line
crosses the y axis
y = 2x - 5
LINE 1
Point-Slope Form of a Linear Equation
1
1
1
1
1 1
1
cross multiply
( ) 1( )
y ym
x x
y ym
x x
m x x y y
y – y1 = m(x – x1)
Where m is the slope of the line and (x1, y1) is a point on the line.
When we know the slope and a point on the line we can use Point-Slope form to determine the equation
Point-Slope Form of a Linear EquationWrite an equation, in slope-intercept form, of a line that goes through the point (-1, 4) and has a slope of 3.
1 1( ) ( )
( 4) 3( ( 1))
4 3( 1)
4 3 3
3 3 4
3 7
y y m x x
y x
y x
y x
y x
y x
Slope intercept form
m = 3
goes through points (-1, 4)
y-intercept (0,7)
Standard Form ax + by = c
-3x + y = 7
3x – y = -7
Point-Slope Form of a Linear EquationWrite an equation, in slope-intercept form, of a line that goes through the point (8, -2) and has a slope of
1 1( ) ( )
3( ( 2)) ( 8)
4
32 6)
4
36 2
4
38
4
y y m x x
y x
y x
y x
y x
3
4
m =
goes through points (8,-2)
y-intercept (0,-8)
Standard Form ax + by = c
- x + y = -8
Multiply by -1
x – y = 8
3
4
3
4
3
4
3 8 246
4 1 4
Slope Intercept Form and Point-Slope Form
Sometimes we may have to use both formulas to find the equation.Find an equation of the line through the points (-1, 5) and (3,-3)Write the equation in slope-intercept form.
3 5 82
3 ( 1) 4
ym
x
1 1( ) ( )
( 5) 2( ( 1))
5 2( 1)
5 2 2
2 2 5
2 3
y y m x x
y x
y x
y x
y x
y x
m = -2
goes through points
(-1, 5) and (3, -3)
y-intercept (0,3)
Standard Form ax + by = c
2x + y = 3
Remember• Positive and negative slopes.
– Positive Slopes move up x number of units and to the right x number of units.
– Negative Slopes move down x number of units and to the right x number of units.
Remember
• If the linear equation does not have a constant term, the y-intercept is the origin (0, 0) y = mx
• Standard Form ax + by = c
• Slope-Intercept Form y = mx + b
• Point-Slope Form y – y1 = m(x – x1)
• Slope1 2
1 2
y yym
x x x
RememberWriting linear equations in slope-intercept form
• If you know the slope and y-intercept form start with slope-intercept form. y = mx + b
• If you know the slope and a point on the line start with point-slope form. y – y1 = m(x – x1)
• If you know two point on the line start by finding the slope
then use the point-slope form. y – y1 = m(x – x1) to find the equation.
1 2
1 2
y yym
x x x
HOMEWORK 7.4
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