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Note-Taking Guides
How to use these documents for success
Print all the pages for the module.
Open the first lesson on the computer.
Fill in the guide as you read.
Do the practice problems on notebook paper (usually).
Put the notes and practice problems in a notebook. You can use these anytime!
Review the notes before you go to sleep. Short term memory is converted to long term
memory ONLY while you sleep. Your brain starts at the end of your day and converts
things to long term memory in reverse order.
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Module #5
5.01 Linear Equations
Look at equations below. Some are linear equations and some are not.
(1) 3x + 5y = 3 is a ______________________ with two variables
(2) 2y = 3x - 6 is a ______________________ with two variables
(3) 2a - 7 = 10 is a ______________________ with one variable
(4) 4x2 + 5 = 7 is ________ a linear equation. Why? __________________________________
(5) xy - 2x = y is ________ a linear equation. Why? __________________________________
(6)
is _______ a linear equation. Why? ____________________________________
From the examples and non examples above, figure out what makes an equation a linear
equation. List the things that linear equations do NOT have. ____________________________
____________________________________________________________________________
____________________________________________________________________________
Now it's your turn to identify linear equations from looking at the equations. Determine if each is
a linear equation. If one is not a linear equation, explain why.
1) x + y = 6
2) xy = 10
3) x2 + y2 = 1
4)
5) x + y =x2
6) x = 0
Don’t forget to check your answers. 
Why are these equations named LINEAR Equations?
You have y = x +3 and in the example, x = 1. Show how to find what y will be. Write down both
steps and label the steps with the words substitute and solve.
If you graph the solution sets of a ____________________, the graph is a straight
__________________.
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In equations such as y = 2x + 1, y is the dependent variable. Why?
____________________________________________________________________________
In the same equations x is the ________________________ variable because
____________________________________________________________________________
Now, go do the practice problems and check your answers. Seek help if you do not understand.
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5.02 Slopes and Intercepts
Slope Intercept Form is
Investigation 1: Understanding the 'm' in the Slope Intercept
Form
Do Investigation 1.
What happens to the line as the value of m increase? _________________________________
What does m stand for in y = mx + b? ______________________________
Making decisions using your knowledge of slope
Answer questions 1 and 2.
Question 1 is 3 because ________________________________________________________
Question 2 is 5 because ________________________________________________________
Investigation 2: Understanding the 'b' in the Slope Intercept Form
Do Investigation 2.
What happens to the line as the value of b changes?
____________________________________________________________________________
What does b stand for in y=mx + b?
____________________________________________________________________________
More about the y intercept
What is the value of x when the line crosses the y axis? ___________
So the ordered pair of the y intercept in y = 3x -2 is (
,
).
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Speaking of Intercepts
What is the value of y when the line crosses the x axis? ___________
Now go to the PRACTICE tab. Find Using your intercept knowledge.
Given the equation 2y = 4x - 6, what are the coordinates of the y intercept and the x intercept?
x=0 at the y intercept
Substitute 0 for x and solve for y. Write the steps below.
The coordinates of the y intercept are (
,
)
y=0 at the x intercept Substitute y=0 in the equation and solve for x. Write the steps below.
The coordinates of the x intercept are (
,
)
Now do the practice problems, check your answers, and seek help if you do not understand.
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5.03 Advance Slope
Method #1 Find the slope of a line if __________________________________
____________________________________________________________________________
Example #1
y= 2x – 3
The slope is _______ because __________________________________________________
Method #2 Find the slope using the graph of a line
m = --------------- = -----------------------------------------Draw a picture of RISE
Draw a picture of RUN
There are 3 steps. Write the steps in your own words.
Step #1 _____________________________________________________________________
Step #2 _____________________________________________________________________
Step #3______________________________________________________________________
Practice Problems
Number 1
Be sure to check your answers!
Number 2
Number 3
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Method #3 Find the slope given 2 points
Write the slope formula here.
m = ----------------------
Copy the example here. Calculate the slope of the line through the points (1, 2) and (3, 5) using
the slope formula. Label the points before you start with x1 y1 and x2 y2.
(1, 2) and (3, 5)
______
______
Now, do the practice problems, check your answers, and seek help if you don’t understand
something.
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5.04 Special Lines
Lesson Part 1
Equations of Horizontal Lines
What is missing from the equation of a horizontal line? _______________________________
Horizontal lines are always expressed as _________________________________________
In the equation y = 3, y is always ___________ no matter what x is.
Where does the equation y = 3 cross the y axis? _______________________
Equations of Vertical Lines
What is missing from the equation of a vertical line?
_______________________________________
Vertical lines are always expressed as
_________________________________________________
In the equation x = -2, x is always ______________ no matter what y is.
Where does the equation x = -2 cross the x axis? _______________________________
Now do Check your understanding and check your answers.
Graph A
1.
2.
3.
4.
Graph B
Graph C
Which graph shows the equation: x = 4?
Which graph shows the equation: y = 0?
Which graph shows the equation: y = -2?
Which graph shows the equation: x = -3?
Tip: Label the graphs with the correct equation for more clarity.
Graph D
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Lesson Part 2
Slopes of Horizontal Lines Read all of this part!
The slope of any horizontal line is __________.
In other words, if a line has a slope of 0, it is a ________________ line.
Slopes of Vertical Lines Read all of this part, too!
You cannot divide by zero so the slope of a vertical line is ___________________________.
The slope of a _______________ line is undefined.
Something to help you remember slopes for horizontal and
vertical lines
H0y Vux stands for
H
0
y
V
u
x
A Simple Jump to Parallel Lines
Parallel lines have ____________________________________________________________
Perpendicular Lines
Perpendicular lines have _______________________________________________________
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5.06 Writing Equations of Lines
To write an equation of a line, you must have _______________ and ____________________.
There are 3 types of problems in this lesson. You will learn…
How to write an equation of a line given_______________________________________
How to write an equation of a line given_______________________________________
How to write an equation of a line given ______________________________________
Writing the equation of a line given the slope and y-intercept
Use ___________________________ where m is the ___________________ and b is the
_________________________
Example 1 Write the equation of a line with the slope of 2 and the y-intercept of (0, -3).
____________________________________________________________________________
____________________________________________________________________________
All you do is __________________________________________________________________
Example 2 Write the equation of a line with a slope of -1/3 and a y-intercept of (0,4)
____________________________________________________________________________
____________________________________________________________________________
Do practice problems 1 and 2 and check your answers.
Practice Problem 1: Write the equation of a line with a slope of 3/4 and a y-intercept of (0,7)
____________________________________________________________________________
Practice Problem 2: Write the equation of a line with a slope of -5 and a y-intercept of (0,-2)
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Write the equation of a line when given the slope and a point on
the line
y=mx + b Method
Copy example 1 here.
Write the equation of a line with a slope of 2 passing through the point (-3, 4).
Copy example 2 here.
Write the equation of a line with a slope of -3 and passing through the point (4, 7).
Point-Slope Formula Method
Copy the point-slope Formula here.
Copy example 1 here.
Write the equation of a line with a slope of 2 passing through the point (-3, 4).
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Copy example 2 here.
Write the equation of a line with a slope of -3 and passing through the point (4, 7).
Practice Problems 2 Choose the method that you like best. Check your answers.
1. Write the equation of a line given m=7 and goes through the point (1, 2).
2. Write the equation of a line given m=2/5 and goes through the point (-5, 4).
3. Write the equation of a line given m=0 and goes through the point (-2, -2).
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Writing the equation of a line given two points on the line
y=mx+b Method
Step 1: ____________________________________________________________
Step 2: ____________________________________________________________
Step 3: ____________________________________________________________
Copy Example: Write the equation of a line that goes through the points (2,3) and (-1, 6).
Does it matter which point you use? _____________________
Point-Slope Formula Method
Copy Example: Write the equation of a line that goes through the points (2,3) and (-1, 6)
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Practice Problems 3
Choose the method that you like best. Check your answers.
1. Write the equation of a line that goes through the points (5,4) and (7,8)
2. Write the equation of a line that goes through the points (-2,-3) and (8,2)
3. Write the equation of a line that goes through the points (1,0) and (0,5)
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5.07 Graphing Linear Equations
Lesson Part 1
In this lesson, you will focus on the following.
______________________________________________________________________
__________________________________________________________
Converting equations written in standard form ____________________________ into
slope-intercept form and graphing them.
Graphing Horizontal and Vertical lines (Remember H0y Vux)
To graph a horizontal line, draw a ________________________ line that intersects the
____axis at the number given in the equation. For example, in y=-2 the line intersects the y axis
at _____________.
To graph a vertical line, draw a _________________________ line that intersects the ____ axis
at the number given in the equation. For example, in x=4, the line intersects the x axis at ____.
Graph examples 1-3 on one graph and 4-6 on the other. Label the lines.
Graphs of Horizontal lines.
1. y = -2
2. y = 3
3. y = 0
Graphs of Vertical Lines
4. x = 4
5. x = 0
6. x = -3
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Graphing Linear Equations in the form y = mx+b
Example: y = ¾ x -2
Step 1: _______________ the slope and y-intercept.
m= ___________________ b = ___________________
Step 2: _______________ the y-intercept. You can think, “begin at b”.
Step 3: Use the __________________ to help graph another point on the line.
Slope is __________ over __________. So the slope of ¾ would mean that you would go
________ 3 units and then to the ___________ 4 units and draw another point.
Lesson Part 2
Changing equations from standard form to slope intercept form.
Standard form: ______________________________
Where A, B and C are ______________________________________________________
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Example 1:
Write the equation in slope intercept
form.
2x + 3y = 6
Copy the rest.
Step 1: (In your own words)
________________________________
________________________________
________________________________
________________________________
________________________________
Step 2:
________________________________
________________________________
Step 3:
________________________________
Graph Example 1. Begin at b, then use the slope to
find the next point.
Example 2:
Write the equation in slope intercept form and then graph it.
4x-2y = 7 Copy the rest.
Now do the Practice Problems and check your answers.
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Graphs for 5.07 Practice Problems
Graph #4, 5 and 6 on the same graph.
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5.11 Solving Systems of Equations
Lesson 1 Graphing Method
When you solve a system of equations, you are looking for
____________________________________________________________________________
____________________________________________________________________________
Solving Systems of Equations Using the Graphing Method
Step 1: Graph x + 2y = 1 Put the equation
in slope-intercept form (y=mx+b form) Write
the steps and enough detail for you to
understand.
Step 2: Graph x – y = 4 Put the equation
in slope-intercept form (y=mx+b form) Write
the steps and enough detail for you to
understand.
Step 3: Find the intersection point. This is the solution to
the system of equations.
Your Turn. Answer the following and check your answers.
1. Would the graphing method always be reliable? Why or why not?
2. What if the 2 lines are parallel? What is the solution?
3. What if the 2 lines are actually the same line? What is the solution? (These type of lines
are called coincidental lines.)
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Practice Using the Graphing Method for Solving System of Equations
Solve each system. Check your answers.
a.
x - y = -3
x+y=9
b.
y = -2x - 6
y = -3x – 10
Lesson 2 - Addition Method
2x – y = 16
x+y=5
The objective of the addition method is to
___________________________________
Copy the steps to solve here. Include the
steps to find y.
___________________________________
___________________________________
Notice that the y terms have
___________________________________
When you add the y terms together you will
get ____________.
To find the y value, substitute __________
___________________________________
What is the solution?
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Your turn
Practice Solving System of Equations Using the Addition Method
Solve each system. Check your answers.
4x - 3y = -10
2x + 3y = 4
2x + 3y = 2
9x - 3y = 42
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5.12 Systems of equations Part 2
Solving Systems of Equations Using Subtraction
When do you use this method? What are the circumstances?
____________________________________________________________________________
Copy the example here. Add enough detail for you to understand each step.
When it says…
“Remember the rules for subtraction: Change the subtraction sign to an addition sign and take
the opposites of the numbers after the sign.”
You can think of this as distributing the negative into the entire equation.
Equation 1: 25x + 16y = 91
Equation 2: 16 x + 16y = 64
Your Turn
Do the practice problems and check your answers.
a. x - 2y = -6
b. 2x +y = -6
x+y=6
3x +y = -10
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Solving systems that use multiplication with the addition or
subtraction
When do you use this method? __________________________________________________
____________________________________________________________________________
Copy the steps for Option 2 here and include enough detail for you to understand the process.
Equation 1: 2x + 5y = 11
Equation 2: 3x - 2y = -12
Do the practice problems. Check your answers.
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5.13 Substitution Method
The object of the substitution method is to _________________________________________
____________________________________________________________________________
Copy the steps to solve the following system of equations here. Include enough detail for you to
understand the process.
a) 2x + y = 11
b) 4x - 3y = 7
Step 1: Solve for one of the variables in one
of the equations. Solve for the variable with
___________________________________
to avoid ____________________________.
Step 2: _______________________ the
value of y (in this case) in to the OTHER
___________________________________
Step 3: Solve the equation.
Step 4: _______________________ your x
value (in this case) into one of the
___________________________________
The solution is ______________________
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Copy the steps to solve the following system of equations here. Include enough detail for you to
understand the process.
Equation 1: x + y = 4
Equation 2: -2x + y = 1
Step 1: Solve for _____________________
___________________________________
Step 2: _______________________ the
value of x (in this case) in to the OTHER
___________________________________
Step 3: Solve the equation.
Step 4: _______________________ your y
value (in this case) into one of the
___________________________________
The solution is ______________________
Do the Practice Problems. Check your answers.
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5.15 Coordinate Geometry Extension
Parallel lines have _____________________________________________________________
Perpendicular lines have ________________________________________________________
Part 1 Parallel Lines
Write the equation of the line that passes through the point (-2, 3) and is parallel to the graph of
y = –2x+ 4. Your final equation should be written in Slope-Intercept Form.
1.
What is the slope of this line? ______________________________________
2. Is the new line parallel or perpendicular to y = -2x + 4? __________________________
3. So the slope of the new line will be ____________________________________
4. Now use point-slope form to write the equation of the new line.
y - y1 = m(x – x1)
Plug in. y1 = _______ m = _________ x1 = ____________ Simplify and solve for y.
Copy the steps here.
Or you can use y=mx + b instead of the point-slope form y - y1 = m(x – x1).
y = ____________, m = ______________, x = ______________
Substitute and solve for b. Copy the steps here.
Substitute back in for _____________________ and _________________.
So the final equations Is __________________________________________
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Part 2 Perpendicular Lines
Write the equation of the line that passes through the point (-2, 3) and is perpendicular to the
graph of y = –2x + 4. Your final equation should be written in Slope-Intercept Form.
1.
What is the slope of this line? ______________________________________________
2. Is the new line parallel or perpendicular to y = -2x + 4? __________________________
3. So the slope of the new line will be __________________________________________
4. Now use point-slope form to write the equation of the new line.
y - y1 = m(x – x1)
Plug in. y1 = _______ m = _________ x1 = ____________ Simplify and solve for y.
Copy the steps here.
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Part 3: Application
In order for this figure to be a true
trapezoid, sides ______________ and
_____________ must be parallel. So the
slopes between points A and B must be
_____________________ the slopes
between points C and D.
Show how to find the slope between A
and B here.
Show how to find the slope between C and D here.
The slopes are ___________________. The lines __________________________ and the
figure ________ a trapezoid.
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Does line segment AD represent the height of the
triangle?
We need to prove that _____________
is ____________________________________ to
_________________ by showing that their slopes
are_____________________________________
Show how to find the slope between A and D here.
Show how to find the slope between B and C here.
The slopes are not ___________________. The lines are not __________________________.
Segment AD does ______________ the height of the triangle.
Do the practice problems and check your answers.