Chapter 5: Linear Equations and Their Graphs

CHAPTER 5: LINEAR EQUATIONS AND THEIR GRAPHS

Notes#27: Sections 5.1 and 5.2

Section 5-1: Rate of Change and Slope

A. Finding rates of change

· 

The rate of change is constant in each table and graph. Find the rate of change. Explain what the rate of change means for each situation.

1.) 2.)

Cost of Renting a Computer
Number of Days / Rental Charge
1 / $60
2 / $75
3 / $90
4 / $105
5 / $120

·  The slope of a line describes its ______and its ______

B. Finding slope

·  Keep in mind that slope of a line is its ______of change

·  Choose and mark two points on the line

·  Count how many steps up (this is the rise of the hill)

·  Count how many steps across (this is the run of the hill)

· 

Lines with a positive slope / Lines with a negative slope / Lines with undefined slope / Lines with zero slope

Find slope of the line. (Double check for +/- )

3.) m = / 4.) m = / 5.) m =
6.) m = / 7.) m = / 8.) m =
Without using a graph and given two points:
and
Slope = m = /

Use the formula to find the slope of the lines containing these points.

Hint: first label points as and

9.) (2, 2) (8, 9) 10.) (-4, -6) (3, -2) 11.) (-2, 3) (2, 1)

12.) (5, -11) (-9, 4) 13.) (9, 7) (3, 7) 14.) (4, -6) (4, 0)

Each pair of points lies on a line with the given slope. Find x or y.

15.) (3, 1), (x, 7) slope = -3 16.) (4, y), (2, 8) slope =
Section 5-2: Slope-Intercept Form

A. Writing linear equations

·  A linear function is a function that graphs a ______.

·  Slope-intercept form of a linear equation is ______, where m stands for ______and b stands for the ______.

·  y-intercept is the coordinate of the point where a line crosses the ______.

·  Example:

Slope = m = ______y-intercept = b = ______

Find the slope (m) and y-intercept (b) of each line.

·  Get y alone. Look for the m and b values in ______.

1.) 3x – 2y = 6 2.) 4x + y = 3 3.) -2x + 3y = -6

4.) x – y = 3 5.) x – 3y = -4

Write an equation of a line with the given slope and y-intercept.

·  Plug m and b into ______

6.) 7.) 8.)

Write the slope-intercept form of the equation for each line.

·  Find the point where the line crosses the y-axis. This is the y-intercept of the line, or ___

·  Count to the next marked point to find the slope of the line, or ____

·  Plug these values into ______

9.)

b = ____, m = ____ / 10.)
b = ____, m = ____ / 11.)
b = ____, m = ____

B. Verifying solutions and graphing linear equations

Determine whether the ordered pair lies on the graph of the given equation.

·  Label your point as (x, y)

·  Plug in both values into the equation. If it is true, then the point is a solution.

12.) (3, 2) 2x – y = 1 13.) (-2, 1) x + 5y = 3

14.) (-4, -2) 3x – 5y = 2 15.) (2, 0)

Lines can be written in either Slope-Intercept form (y = mx + b) or Standard Form (Ax + By = C). You need to know how to convert from one to the other.

Converting to Slope-Intercept Form
Goal: y = mx + b
(where m and b are integers or fractions)
·  Get y alone
·  Reduce all fractions / Converting to Standard Form
Goal: Ax + By = C
(where A, B, and C are integers and
where A is positive)
·  Get x and y terms on the left side and the constant term on the right side of the equation
·  Multiply ALL terms by the common denominator to eliminate the fractions
·  If necessary, change ALL signs so that the x term is positive
16.) Convert to slope-intercept form:
4x – 12y = 8 / 17.) Convert to standard form:

18.) Convert to both slope-intercept form and standard form:
a) y – 3 = -5(x + 4) b.)

Notes #28: Sections 5.3 and 5.4

A. Graphing Lines using the slope and y-intercept:
- Get y alone so the equation is in y = mx + b form (m = ______, b = ______)
- Graph b first. This point goes on the ____ axis.
- Use slope and count rise over run to the next point(s). When you have at least three points on your graph, then connect the points with a ruler to make a straight line.
- Label your graphed line with the original equation
Most common errors:
·  Graphing b on the x-axis instead of the y-axis
·  Graphing the slope in the wrong direction (e.g. forgetting a negative)
1.)
(I’m already in slope-intercept form!)
m = ___ (ß graph me second! Watch the negative!)
b = ___ (ß graph me first! I go on the y-axis!)
2.) x – 2y =2
(Get me in slope-intercept form first)
m = ______
b = ______
3.) x + 3y = -6
(Get me in slope-intercept form first)
m = ______
b = ______/

Section 5-3: Standard Form

A. Graphing Equations Using Intercepts

·  x-intercept is the x-coordinate of the point where a line crosses the ______. To find the x-intercept, make y = 0 and solve for x.

·  y-intercept is the y-coordinate of the point where a line crosses the ______. To find the y-intercept, make x = 0 and solve for y.

Find the x- and y-intercepts.

1.) 2.)

x-intercept y-intercept

(make y = 0) (make x = 0)

(____, 0) and (0, ____)

Graphing Lines using the x- and y- intercepts. The intercepts are the point(s) where a line intersects the axes of the coordinate plane.
- Find the x and y intercepts (by setting the opposite variable to zero)
- Write these answers as two different points
- Graph and connect these points to graph the line
- Label the graphed line with the original equation
Most common error:
Forgetting that the intercepts are two different points and graphing as just one
3.) x + 2y = 4
x-intercept y-intercept
(set y = 0) (set x = 0)
x-int: ( , 0) y-int: (0, ) /
4.) 3x – y = 3
x-int: ( , ) y-int: ( , ) /
5.) 2x – 3y = 8
x-int: ( , ) y-int: ( , ) /
Special Cases: Graphing Horizontal and Vertical Lines
6.) x = 4
(This equation describes the line for which ALL points have an x-coordinate of 4. There are no restrictions on the value of y). / 7.) y = -2
(This equation describes the line for which ALL points have a y-coordinate of -2. There are no restrictions on the value of x).
Use the pattern you found above to complete these sentences:
• Any line in the form x = ___ is a ______line because it intersects the ______
• Any line in the form y = ___ is a ______line because it intersects the ______
Use this pattern to graph these lines without a table of solutions.
8.) y = 3 / 9.) x = -2 / 10.) y = -4

Write an equation in standard form to describe each situation. Be sure to define your variables.

11.) Two apples and three bananas cost a total of $1.60. Seven apples and four bananas cost $4.30.

12.) A free lance photographer makes $50 per photograph that is published in the newspaper and $100 per photograph that is published in a magazine. The photographer needs to earn $700. Write an equation describing the photographer’s next week.

13.) Write an equation in standard form to find the number of minutes someone who weights 150 lb would need to bicycle and swim laps in order to burn 300 Calories. Use the fact that a 150 lb person burns 10 Calories per minute riding a bike and 12 Calories per minute swimming laps.

Notes #29: Section 5.4

Section 5-4: Point-Slope and Writing Linear Equations

A. Point Slope Form

·  Point-Slope form of the equation of a non-vertical line that passes through point and has slope m is:

·  Your textbook emphasizes point-slope form, but we will not use it in this class. If an equation is given in point slope form, always convert it into y = mx + b form.

Graph each equation.

1.)
(Get me in slope-intercept form first) /
2.) /

B. Writing linear equations given the slope and y-intercept

- Find the slope (m) and y-intercept (b) [If the given information is a graph, then you will have to count by hand to find these values.]

- Fill in m and b so you have an equation of the line in y = mx + b form.

y = ______x + ______

(Put m here!) (Put b here!)

3.) Find the equation of the line with slope of 5 and y-intercept
of -2. Write in standard form. / 4.) Find the equation of the given line in slope-intercept form.
/ 5.) Write the equation of a line that has the same slope as and has a y-intercept of 1. Write in standard form.

C. Writing linear equations given the slope and a point

·  plug slope = m into y = mx + b
·  name your point (x, y) and plug these values in for x and y
·  solve for b
·  plug m and b back into y = mx + b
·  convert to standard form, if necessary
** Remember to leave x and y as variables! **
6.) Find the equation of the line with slope of -2 and going through (-1, 3) in slope-intercept form. / 7.) Find the equation of the line with slope of and going through (6, -2) in standard form.
8.) Find the equation of the line in slope-intercept form with slope and passing through the point (-3, 7). / 9.) Find the equation of the line with the slope of zero going through the point (-4, 5) in standard form.

D. Writing linear equations given two points

·  find the slope using the slope formula
·  pick one of your points to be x and y
·  plug m, x, y into y = mx + b
·  solve for b; plug m and b into y = mx + b
·  convert to standard form, if necessary
** Remember to leave x and y as variables! **
10.) Find the equation of the line going through (-3, 1) and (4, 8) in slope-intercept form.
/ 11.) Find the equation of the line with
x-intercept 3 and y-intercept -2 in standard form.
12.) Find the equation of the line going through (5, 2) and (-1, 3) in standard form. / 13.) Find the equation of the line with
x-intercept 5 and y-intercept -4 in slope-intercept form.

Notes #30: Review

Graphing Lines using the x- and y- intercepts
Find the x and y intercepts (by setting the opposite variable to zero). These answers represent two different points. Use these points to graph the line.
1.) x + 2y = 4
x-int: ( , ) y-int: ( , ) / 2.) 3x – y = 3
x-int: ( , ) y-int: ( , )
Graphing Lines using the slope and y-intercept:
1. Get the equation into y = mx + b form
2. Graph b first. This point goes on the ____ axis.
3. Use slope and count rise over run to the next point(s). Watch for negative slope!
/
3.) x – 2y =2 / 4.) x + 3y = -6
Vertical and Horizontal Lines
Graph each line.
5.) x = 2 / 6.) y = -3
Writing Linear Equations
Find the slope and y-intercept of each line and write an equation of the line in y = mx + b form.
y = ______x + ______
7.
/ 8.

9. Find the equation of the line with slope of 5 and y-intercept of -2. Write in slope-intercept form. / 10.) Find the equation of the line with slope of and y-intercept of -1. Write in standard form.
11. Find the equation of the line with slope of -2 and going through (-1, 3). Write in slope-intercept form. / 12. Find the equation of the line with slope of and going through (6, -2).
13. Find the equation of the line with slope of and going through (1, 5). / 14. Find the equation of the line going through
(-1, 0) and (4, 2) in slope-intercept form.
15. Find the equation of the line with
x-intercept -1 and y-intercept 2 in standard form. / 16. Find the equation of the line going through
(-3, 4) and (-3, 7) in standard form.

Notes#31:

Section 5-5: Perpendicular and Parallel Lines

A. Determining if lines are parallel or perpendicular

Draw two parallel lines: Draw two perpendicular lines:

What can you say about their slopes? What can you say about their slopes?

Parallel lines have ______slope. Perpendicular lines have ______

______slope.

Examples: Examples:

The slope of a line is given. Find the slope of a line parallel to it and a line perpendicular to it.