Solving Equations

1
Chapter 6
Algebra I
Algebra I

• 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
2
Rate of Change Slope
Lesson 6-1

• 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 Slope vertical
change rise horizontal
change run Finding Slope using a
graph Slope 3 - 1 4 (-1)
2/5
Choose two points 0 - 1000 180 - 60 -1000/ 120
- 25/3 or -8 1/3
3
Rate of Change Slope
Lesson 6-1

• Notes

Your turn Find the slope -1/6 Finding slope
using points Slope rise y2 y1 , where
x2 x1 ? 0 run x2 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.
4
Rate of Change Slope
Lesson 6-1

• Notes

Find the slope Choose two points
Find the slope Choose two points
5
Rate of Change Slope
Lesson 6-1

• Homework

Homework Practice 6-1 even
6
Slope-Intercept Form
Lesson 6-2

• Practice 6-1

7
Slope-Intercept Form
Lesson 6-2

• 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
8
Slope-Intercept Form
Lesson 6-2

• 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
9
Slope-Intercept Form
Lesson 6-2

• Homework

Homework Practice 6-2 odd
10
Standard Form

• Practice 6-2

Lesson 6-3
11
Standard Form

• Practice 6-2

Lesson 6-3
12
Standard Form
Lesson 6-3

• 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
13
Standard Form
Lesson 6-3

• 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)
14
Standard Form
Lesson 6-3

• 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)
15
Standard Form
Lesson 6-3

• Homework

Homework Practice 6-3 odd
16
Point-Slope Form Writing Linear Eq.
Lesson 6-4

• Practice 6-3

17
Point-Slope Form Writing Linear Eq.
Lesson 6-4

• Practice 6-3

18
Point-Slope Form Writing Linear Eq.
Lesson 6-4

• 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)
19
Point-Slope Form Writing Linear Eq.

• Notes

Lesson 6-4
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)
20
Point-Slope Form Writing Linear Eq.
Lesson 6-4

• Homework

Homework Practice 6-4 odd
21
Parallel Perpendicular Lines
Lesson 6-5

• Practice 6-4

22
Parallel Perpendicular Lines
Lesson 6-5

• Practice 6-4

23
Point-Slope Form Writing Linear Eq.

• Notes

Lesson 6-4
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 ? In Summary
Slope-Intercept form y mx b Standard
Form Ax By C Point-Slope Form (y y1)
m (x x1)
x y
-1 4
3 6
5 7
11 10
x y
-11 -7
-1 -3
4 -1
19 5
24
Parallel Perpendicular Lines
Lesson 6-5

• 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
¾ x 3 y ¾ x 7 parallel? The lines
are parallel. The equations have the same slope,
¾, different y-intercepts. Writing equations of
parallel lines Given a point and an equation,
write the equation for the parallel line passing
through the given point.
25
Parallel Perpendicular Lines
Lesson 6-5

• Practice 6-4

26
Parallel Perpendicular Lines
Lesson 6-5

• 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 lines slope will
be the negative reciprocal) Slope -2/5
Perpendicular slope ?
27
Parallel Perpendicular Lines
Lesson 6-5

• 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
28
Parallel Perpendicular Lines

• Homework

Lesson 6-5
Homework Practice 6-5 13-36
29
Scatter Plots Equations of Lines
Lesson 6-6

• Practice 6-5

30
Scatter Plots Equations of Lines
Lesson 6-6

• Practice 6-5

31
Graphing Absolute Value Equations
Lesson 6-7

• 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
32
Graphing Absolute Value Equations
Lesson 6-7

• Notes

Writing an Absolute Value Equation Write an
equation for each translation of y x 8 units
down 6 units up
2 units up 5 units down y x - 8
y x 6
y x 2 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?
33
Graphing Absolute Value Equations
Lesson 6-7

• Notes

Writing an Absolute Value Equation Write an
equation for each translation of y x 8 units
left 6 units right 5
unit right 7 units left y x 8
y x -6 y x - 5 y x
7
34
Graphing Absolute Value Equations
Lesson 6-7

• Homework

Homework Practice 6-7 odd
35
Graphing Absolute Value Equations
Lesson 6-7

• Practice 6-6

36
Graphing Absolute Value Equations
Lesson 6-7

• Practice 6-7

37
Graphing Absolute Value Equations
Lesson 6-7

• Practice 6-7

38
Chapter 6
Algebra I
Algebra I

• Chapter Review

39
Chapter 6
Algebra I
Algebra I

• Chapter Review