~ Chapter 6 ~
Solving Equations
Lesson 6-1 Rate of Change & Slope
Lesson 6-2 Slope-Intercept Form
Lesson 6-3 Standard Form
Lesson 6-4 Point-Slope Form & Writing Linear Eq.
Lesson 6-5 Parallel & Perpendicular Lines
Lesson 6-6 Scatter Plots & Equations of Lines
Lesson 6-7 Graphing Absolute Value Equations
Chapter Review
Rate of Change & Slope
Notes
Rate of change – change in the dependent variable or vertical change
change in the independent variable
horizontal change
Find the rate of change…
120 - 60
5-1
60/4 = 15/1
Choose two
points
0 - 1000
180 - 60
-1000/ 120 =
- 25/3 or -8 1/3
Slope = vertical change
horizontal change
= rise
run
Finding Slope using a graph…
Slope = 3 - 1
4 – (-1)
2/5
Rate of Change & Slope
Notes
Your turn…
Find the slope…
-1/6
Finding slope using points…
Slope = rise = y2 – y1 , where x2 – x1 ≠ 0
run
x 2 – x1
C (2,5) & D (4,7)
So… Slope = 7 - 5
4–2
2/2 = 1/1 = 1
Your turn… P(-1,4) & Q(3,-2)
Slope = -6/4 = -3/2
Horizontal lines have a slope of 0. Vertical lines have a slope that is undefined.
Rate of Change & Slope
Notes
Find the slope…
Choose two points
Find the slope…
Choose two points
Rate of Change & Slope
Homework
Homework – Practice 6-1 even
Slope-Intercept Form
Practice 6-1
Slope-Intercept Form
Notes
Writing Linear Equations
Linear equation – an equation whose graph is a line. (ex: y = 2x – 3)
y-intercept is the y coordinate of the point where the line crosses the y-axis.
If you know the slope of a line (use the slope formula) and the y-intercept…
you can write a linear equation.
Write the linear equation…
Slope = 1 – 4 = -3/8
8–0
So… y = -3/8x + 4
The slope-intercept form of a Linear equation
Find the slope and y-intercept of each equation…
y = -2x + 1
y = 7/6 x – ¾
y = -4/5 x
If you know the slope ( -3) and the y-intercept (4), write the linear equation…
y = -3 x + 4
Slope-Intercept Form
Notes
Writing an Equation from a graph…
Find the slope… (Choose 2 points on the line)
Slope = 1 – 2 = -1/-2 = 1/2
0–2
y = 1/2 x + 1
Graphing Equations
Step 1 – Plot the y-intercept point.
(0, y-intercept)
Step 2 – Use the slope to plot the 2nd point.
Step 3 – Draw a line through the two points.
Graph y = 2x – 1
Graph y = 3/2 x - 2
Slope-Intercept Form
Homework
Homework ~ Practice 6-2 odd
Standard Form
Practice 6-2
Standard Form
Practice 6-2
Standard Form
Notes
Standard form of a linear equation… Ax + By = C, where A, B, & C are real
numbers and A & B are not both zero.
Finding the x- & y-intercepts
The x-intercept is the point on the line where y is zero, or (x-intercept, 0), so…
In 3x + 4y = 8, solve for x using the y value of 0
3x + 4(0) = 8
3x = 8
x = 8/3, or 2 2/3 so (8/3, 0) is where the line crosses the x-axis.
The y-intercept is the point on the line where x is zero, or (0, y-intercept), so…
In 3x + 4y = 8, solve for y using the x value of 0
3(0) + 4y = 8
4y = 8
y = 8/4 = 2, so (0, 2) is where the line crosses the y-axis.
Your turn… find the x- & y-intercepts for 4x – 9y = -12
Standard Form
Notes
The x-intercept = -3 & the y-intercept = 4/3
Graphing using the x- & y-intercepts
Step 1 – Find the x- & y-intercepts
Step 2 – Plot the points (x-intercept, 0) & (0, y-intercept)
Step 3 – Connect the dots… Draw a line through the points.
Graph 5x + 2y = -10
Graphing horizontal & vertical lines…
y = -3 … write in standard form…
0x + 1y = -3, so for all values of x, y = -3
x = -4… write in standard form…
1x + 0y = -4, so for all values of y, x = -4
Transforming to standard form
Write -2/5x +1 in standard form (Ax + By = C)
Standard Form
Notes
5y = 5(-2/5 x + 1)
5y = -2x + 5
2x + 5y = 5 (standard form)
You try… y= 2/3x + 5
-2x + 3y = 15 (standard form)
y = -4/5x – 7
4x + 5 y = -35 (standard form)
Standard Form
Homework
Homework – Practice 6-3 odd
Point-Slope Form & Writing Linear Eq.
Practice 6-3
Point-Slope Form & Writing Linear Eq.
Practice 6-3
Point-Slope Form & Writing Linear Eq.
Notes
Using Point-Slope form… for a Linear Equation
The point-slope form of the equation of a non vertical line that passes
through the point (x1, y1) with slope of m…
y – y1 = m (x – x1)
So, write the point slope form of the equation of a line through the point
(3, 4) and with slope of 2
y – 4 = 2 (x – 3)
Your turn…
slope of 2/5 and passes through the point (10, -8)
y – (-8) = 2/5 (x - 10) or y + 8 = 2/5 (x – 10)
Graphing using the point-slope form
Step 1 – Plot the point the equation shows.
Step 2 – Plot the next point using the slope.
Step 3 – Connect the dots.
Graph the equation y – 5 = ½ (x – 2)
Point-Slope Form & Writing Linear Eq.
Notes
Graph y – 5 = -2/3(x + 2)
Using 2 Points to Write an Equation
Write equations for a line in point-slope form and in slope-intercept form.
Step 1 - Choose two points and find the slope
Step 2 – Use either point used to find the slope to write the equation in
point-slope form.
Step 3 – Rewrite the equation from Step 2 in slope-intercept form.
Slope = -5 – 3 = -8/-3 = 8/3
-1 – 2
y – (-5) = 8/3 (x – (-1)) =
y + 5 = 8/3 (x + 1) (point-slope form)
y + 5 = 8/3x + 8/3
y = 8/3x – 2 1/3 (slope-intercept form)
Point-Slope Form &
Writing Linear Eq.
Homework
Homework – Practice 6-4 odd
Parallel & Perpendicular Lines
Practice 6-4
Parallel & Perpendicular Lines
Practice 6-4
Point-Slope Form & Writing Linear Eq.
Notes
Writing an Equation Using a Table
Step 1 – Find the rate of change for consecutive ordered pairs. (Determine
if the relationship is linear. Hint: each rate of change is same for each
camparison)
Step 2 – Use the slope & a point to write an equation.
Rate of change for each is ½
So y – 6 = ½ (x - 3)
What about 
x
y
-11
-7
In Summary ~
Slope-Intercept form:-1y = mx +-3
b
Standard Form: Ax + By = C
4
-1
Point-Slope Form: (y – y1) = m (x – x1)
19
5
x
-1
3
5
11
y
4
6
7
10
Parallel & Perpendicular Lines
Notes
Slopes of parallel lines – nonvertical lines are parallel if they have the same
slope and different y-intercepts. For example ~ y = ½ x + 3 & y = ½ x – 1
have the same slope of ½ and different y-intercepts so they are parallel.
Determine whether lines are parallel…
Step 1 – solve both equations for y (slope-intercept form)
Step 2 – compare slope and y-intercepts.
Determine if -6x + 8y = -24 & y = ¾ x – 7 are parallel.
8y = 6 x – 24
y = 6/8 x – 3
y=¾x–3 &y=¾x–7
parallel?
The lines are parallel. The equations have the same slope, ¾, & different yintercepts.
Writing equations of parallel lines
Given a point and an equation, write the equation for the parallel line passing
through the given point.
Parallel & Perpendicular Lines
Practice 6-4
Parallel & Perpendicular Lines
Notes
Step 1 – Identify the slope of the given line.
Step 2 – Use the given point to write the point-slope form equation and then
convert to the slope-intercept form.
Write an equation for the line that contains (2, -6) and is parallel to
y = 3x + 9
Slope = 3
y – (-6) = 3 (x – 2) (solve for y)
y + 6 = 3x – 6
y = 3x – 12
Slopes of Perpendicular lines – Two lines are perpendicular if the product
of their slopes is -1. A vertical and a horizontal line are also perpendicular.
For example: slope is ¾ so the perpendicular line would have a slope of -4/3.
(Hint: The perpendicular line’s slope will be the negative reciprocal)
Slope = -2/5
Perpendicular slope = ?
Parallel & Perpendicular Lines
Notes
Writing Equations for Perpendicular Lines
Given a point and an equation of a line, find the equation of a line
perpendicular to the given line.
Step 1 – Identify the slope of the given line.
Step 2 – Convert the slope (negative reciprocal)
Step 3 – Write the point-slope form of the equation and convert to the
slope-intercept form of the equation.
Write an equation of the line that contains (1, 8) and is perpendicular to
y = ¾ x + 1.
Slope = ¾
Negative reciprocal = -4/3
y – 8 = -4/3(x – 1)
y – 8 = -4/3x + 4/3
y = -4/3x + 9 1/3
Parallel & Perpendicular Lines
Homework
Homework ~ Practice 6-5
#13-36
Scatter Plots & Equations of Lines
Practice 6-5
Scatter Plots & Equations of Lines
Practice 6-5
Graphing Absolute Value Equations
Notes
Absolute value equation – V-shaped graph pointing upward or downward.
Translation – shift of a graph horizontally, vertically, or both. (slide)
Vertical Translations
y = |x|
So, what would y = |x| + 2 look like?
What about y = |x| - 3?
So, the graph of y = |x| + k is a translation of y = |x|. If k is positive then
the graph translates up k units, and if k is negative then the graph translates
down k units.
To show a vertical translation… graph y = |x| and then graph the translation
y = |x| + k
Graphing Absolute Value Equations
Notes
Writing an Absolute Value Equation
Write an equation for each translation of y = |x|…
8 units down
6 units up
y = |x| - 8
y = |x| + 6
2 units up
y = |x| + 2
5 units down
y = |x| - 5
Horizontal Translations
y = |x + h| is a translation of y = |x|. If h is positive then the graph
translates h units to the left. If h is negative then the graph translates h
units to the right.
Graph the equation by translating y = |x| for y = |x - 2|
What about y = |x + 2|?
What about y = |x - 4|?
Graphing Absolute Value Equations
Notes
Writing an Absolute Value Equation
Write an equation for each translation of y = |x|…
8 units left
6 units right
y = |x + 8|
y = |x -6|
5 unit right
7 units left
y = |x - 5|
y = |x + 7|
Graphing Absolute Value
Equations
Homework
Homework – Practice 6-7 odd
Graphing Absolute Value Equations
Practice 6-6
Graphing Absolute Value Equations
Practice 6-7
Graphing Absolute Value Equations
Practice 6-7
~ Chapter 6 ~
Chapter Review
~ Chapter 6 ~
Chapter Review