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Unit 2 – Solving Equations 1
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– Unit 3a – [Graphing Linear Equations]
Unit 2 – Solving Equations 2
To be a Successful Algebra class,
TIGERs will show…
#TENACITY during our practice, have…
I attempt all practice I attempt all homework I never give up when I don’t understand
#INTEGRITY as we help others with their work, maintain a…
I always check my answers I correct my work, I never just copy answers I explain answers, I never just give them
#GO-FOR-IT attitude, continually…
I write down all notes, even if I’m confused I remain positive about my goals I treat each day as a chance to reset
#ENCOURAGE each other to succeed as a team, and always…
I offer help when I understand the material I push my teammates to reach their goals I never let my teammates give up
#REACH-OUT and ask for help when we need it!
I ask my questions during homework check I ask my teammates for help during practice I attend enrichment/tutorials when I need to
Unit 2 – Solving Equations 3
Unit Calendar
MONDAY TUESDAY WEDNESDAY THURSDAY FRIDAY
October 6 7 8 9 10
Graphing from a
Table
Slope
Slope
QUIZ
Slope Intercept
Form
Parent Function
And Transformations
13 14 15 16 17
Student Holiday Slope Intercept
Form
QUIZ
PSAT Standard Form and
Intercepts
Standard to Slope
Intercept Form
20 21 22 23 24
Solving for Slope
Intercept Form
Review
TEST
…
…
Essential Questions
What does it mean for relationship to be a linear function?
How can linear functions be used to model problem situations?
What are the critical features of a graph and how can they help me solve this problem?
How can a value of a function be represented?
What is the meaning of slope and how do I find it in this situation?
Unit 2 – Solving Equations 4
Critical Vocabulary
Linear Function
Parent Function
Slope
x-Intercept / Zero
y-intercept
Direct Variation
Parallel
Perpendicular
Unit 2 – Solving Equations 5
Graphing from a Table
Examples:
Equation: 𝑦 = 3𝑥 – 5 Table Graph
x y
-2
-1 0
1 2
3 . . .
. . .
Equation: 𝑦 = −2𝑥 + 6
Table Graph
x y
-2
-1 0
1
2 3 . . .
. . .
To babysit my younger brother, my mom gives me $5 and an extra $2 per hour.
Equation: 𝑦 =
Table Graph x y
-2 -1
0
1 2
3 . . .
. . .
Equation: 𝑦 = −2
3𝑥 + 5
Table Graph x y
-6 -3
0
3 6
9 . . .
. . .
Unit 2 – Solving Equations 6
Practice:
Equation: 𝑦 = 2𝑥 – 4
x y
-2
-1 0
1 2
3 . . .
. . .
Equation: 𝑦 = −4𝑥 + 3
x y
-2
-1
0 1
2 3 . . .
. . .
With no money in my bank, I started saving $3 every week.
Equation: 𝑦 =
x y
-2 -1
0
1 2
3 . . .
. . .
My box of chocolates had 5 pieces in it. I ate one piece each class period.
Equation: 𝑦 =
x y
-2 -1
0
1 2
3 . . .
. . .
Equation: 𝑦 =1
3𝑥– 4
x y
-6 -3
0 3
6 9 . . .
. . .
Equation: 𝑦 = −3
2𝑥 + 4
x y
-4
-2 0
2 4
6 . . .
. . .
Unit 2 – Solving Equations 7
Unit 2 – Solving Equations 8
Slope
Slope: The slope of a line is the measure of how ________ the line is. Slope is directly related to the ______ ___ _______ in a rule.
Slope Formula: m = 𝒚𝟐 − 𝒚𝟏
𝒙𝟐 − 𝒙𝟏
𝑪𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒚
𝑪𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒙
𝑹𝒊𝒔𝒆
𝑹𝒖𝒏
Positive Negative Zero (0) Undefined
Increases from Left to Right
Decreases from Left to Right
Stays the same (constant) from Left to Right
Not a function, so no definition for slope
Unit 2 – Solving Equations 9
Examples: Slope Formula: m = 𝒚𝟐 − 𝒚𝟏
𝒙𝟐 − 𝒙𝟏
Table 2 - Points Graph Situation
x y -2 12
-1 10
0 8 1 6
2 4
( -2 , 12 ) and ( 1 , 6 )
I have $8 to spend on snacks during the week. Each snack I purchase costs $2.
x y
-4 -1 -2 2
0 5 2 8
4 11
( 0 , 5 ) and (-2 , 2 )
Every 2 months I have to pay a $3 charge to keep my account open. Originally, I had to pay a $5 fee to open the account.
x y
-2 4 -1 4
0 4 1 4
2 4
( -1 , 4 ) and (2 , 4 )
A solar powered go cart began moving at 4 mph and never changed speed.
x y
-3 -2 -3 -1
-3 0
-3 1 -3 2
( -3 , -2 ) and ( -3 , 1 )
N/A This is not a function, so it does not have a situation to match it.
Unit 2 – Solving Equations 10
Practice:
x y
-4 -14
-2 -7 0 0
2 7 4 14
m =
( 1 , 3 ) and ( 7 , 6 )
m =
m =
I am paying my friend back $4 every week from the $20 I borrowed from her.
m =
m =
m =
x Y
1 -2 2 -2
3 -2 4 -2
5 -2
( 0 , 0 ) and ( 9 , -6 )
m =
m =
x y
-2 -3 -1 2
0 7 1 12
2 17
Benjamin got a $40 gift card to Starbucks for his birthday. His favorite drink costs $3 each time he goes. m =
( 4 , -1 ) and ( 4 , 9 ) m =
Every 6 months I pay my car insurance $131, plus I paid an intial fee of $25 when I signed up. m =
m =
x y -3 -4
-2 -3 -1 -2
0 -1
1 0
m =
m =
( 4 , -5 ) and ( -2 , -7 )
m =
m =
x y
-5 17
-3 11
-1 5
1 -1 3 -7
Unit 2 – Solving Equations 11
Unit 2 – Solving Equations 12
Slope Intercept Form
Slope-Intercept Form: The equation of a line written in a form that includes both the ________ and the ___ - _____________.
(m) (b)
Examples:
𝑦 = 3𝑥– 5
𝑦 = − 𝑥 + 7
𝑦 = 4𝑥
𝑦 =1
4𝑥 − 5
𝑦 = −3
2𝑥 + 7
𝑦 =2
5𝑥
Unit 2 – Solving Equations 13
Special Cases: Horizontal and Vertical Lines
H O Y
Horizontal Line
Zero Slope
0
5
𝑦 = constant
𝑦 = 2
V
Vertical Line
U
Undefined Slope
3
0
X
𝑥 = constant
𝑥 = −4
Examples:
𝑦 = −2
𝑥 = −2
𝑦 = −2𝑥
Unit 2 – Solving Equations 14
Practice:
𝑦 = 2𝑥 + 3
𝑦 =2
3𝑥 + 1
𝑦 = 6
𝑦 = −3𝑥
𝑦 = 𝑥– 5
𝑦 = −4
3𝑥– 1
𝑥 = 7
𝑦 = −3
2𝑥
𝑦 = −4𝑥 + 9
Unit 2 – Solving Equations 15
Linear Parent Function and Transformations
Transformations: The parts of an equation that change the shape or position of a graph/line.
Slope / Steepness: Ignoring the negative, if the slope is greater than one then the line is steeper.
𝑦 = 𝑥 𝑦 = 4𝑥 𝑦 =
5
2𝑥
**Improper Fractions Ignoring the negative, if the slope is less than one then the line is less steep.
𝑦 = 𝑥 𝑦 =
1
2𝑥
**Proper Fractions
Unit 2 – Solving Equations 16
Tran SL ation: If there is a constant (regular number with no variable), SL ide the change is a vertical shift or slide of the line.
𝑦 =
1
2𝑥 𝑦 =
1
2𝑥 + 3
Re FL ection: If the coefficient (number in front) of x is negative,
FL ip the change is a vertical flip of the line.
𝑦 =
1
2𝑥 + 3 𝑦 = −
1
2𝑥 + 3
Parent Function: The most basic form of a graph (No Transformations)
Linear Parent Function: 𝑦 = 𝑥 “𝑦 = +1𝑥 + 0”
No Reflection, Slope of 1, No Translation
Unit 2 – Solving Equations 17
Examples:
𝑦 = 4𝑥 − 3
Steepness: More Steep Less Steep No Change
Translation: Shift Up Shift Down No Change
Reflection: Reflection No Change
Parent Function
Transformed Equation
𝑦 = −
2
5𝑥 + 1
Steepness: More Steep Less Steep No Change
Translation: Shift Up Shift Down No Change
Reflection: Reflection No Change
Parent Function
Transformed Equation
Equation Steepness Translation Reflection
𝑦 =4
3𝑥 + 5
𝑦 = −3𝑥
𝑦 = −𝑥 − 27
Unit 2 – Solving Equations 18
Practice:
𝑦 =
4
3𝑥 + 1
Steepness: More Steep Less Steep No Change
Translation: Shift Up Shift Down No Change
Refelction: Reflection No Change
Parent Function
Transformed Equation
Equation Steepness Translation Reflection
𝑦 = −6𝑥 + 5
𝑦 =2
7𝑥 − 1
𝑦 = −𝑥
𝑦 = −3𝑥 − 8
𝑦 =3
2𝑥 − 6
𝑦 = 𝑥 + 4
𝑦 =1
4𝑥
𝑦 = 𝑥 − 35
Unit 2 – Solving Equations 19
Unit 2 – Solving Equations 20
Slope Intercept Form (Applications)
Examples: I owe my friend $10 and I plan to pay them back $2 each day. The equation that models this situation is
𝑦 = 2𝑥 – 10
A) What does the y represent?
B) What does the x represent?
C) What does the 2 represent?
D) What does the 10 represent?
E) Why is the 10 negative in this problem?
F) Would this be continuous or discrete?
G) Graph the equation
H) What is a reasonable domain for this problem?
Gina has a $30 gift card to Starbuck’s. Every time she buys a coffee she is charged $3.
A) Write an equation in Slope-Intercept Form to represent the problem
B) Why did you choose the sign of 3 to be positive or negative?
C) If we doubled the constant in the equation, what would that represent in the problem?
D) What is a reasonable domain for this situation?
E) Is this situation discrete or continuous?
Unit 2 – Solving Equations 21
Practice:
Chris does a lot of babysitting. When parents drop off their children and Chris can supervise at home, the
hourly rate is $3. If Chris has to travel to the child’s home, there is a fixed charge of $5 for transportation in
addition to the $3 hourly rate.
A) Write an equation for when he watches the
kids at home?
B) Write an equation for when he needs to travel to watch the kids?
C) Graph both Lines
D) Will a parent dropping off their kid ever pay more than if Chris has to travel to their home?
E) If in 3 years, Chris has a new equation, y = 5x + 10, what has changed in this problem?
The charges for Anderson’s Plumbing can be modeled by the following equation, where, C, is the total cost for
plumbing services that last, h, hours:
40 25C h
A) What does the 25 represent in this situation? B) What does the 40 represent in this situation?
The cost of dinner at a sweet 16 party is $300 plus $10 per person.
A) What is the equation that models this situation?
B) What does the y-intercept represent? C) What does the slope represent?
D) Find the cost for 50 guests.
Unit 2 – Solving Equations 22
Laura is on a 2 day hike in the Smoky Mountains. She hiked 8 miles the first day and is hiking at a rate of
3 mi/h on the second day. Her total distance is a function of the time she hikes.
A) What is the equation that models this situation?
B) What does the y-intercept represent? C) What does the slope represent?
D) What will be Laura’s total distance if she hikes 6 hours on the second day?
The equation for a car driving away from Katy is represented by the equation: 46y x , where y is the
distance from Katy after x hours.
A) What does y represent in this problem?
B) What does x represent in this problem?
C) What is the meaning of the slope?
D) What would the slope and y-intercept in the equation 46 184y x represent?
The Star Car Rental Company charges a flat fee of $30 plus $0.25 per mile to rent a car.
A) Which value would be considered the y-intercept or beginning value?
B) Which value would be considered the slope or rate of change?
C) Write the equation for this situation.
D) If You only had $50 to spend on the rental car, how many miles could you drive?
Unit 2 – Solving Equations 23
Unit 2 – Solving Equations 24
Standard Form and Intercepts
Standard Form: The equation of a line written in the form Ax + By = C A is always positive and C is a constant. x - intercept: The point where a graph touches the x – axis ( ___ , 0 ) y - intercept: The point where a graph touches the y – axis ( 0 , ___ ) Examples:
Equation: 3x + 5y = 30 x – intercept: ( , 0 ) y – intercept: ( 0 , ) Slope:
Equation: 4x – 6y = -12 x – intercept: ( , ) y – intercept: ( , ) Slope:
Unit 2 – Solving Equations 25
Practice:
Equation: 2x + 8y = -16 x – intercept: ( , ) y – intercept: ( , ) Slope:
Equation: 5x – 10y = 20 x – intercept: ( , ) y – intercept: ( , ) Slope:
Equation: 9x + 3y = 18 x – intercept: ( , ) y – intercept: ( , ) Slope:
Equation: 10x – 5y = -30 x – intercept: ( , ) y – intercept: ( , ) Slope:
Equation: 4x + 6y = -24 x – intercept: ( , ) y – intercept: ( , ) Slope:
Equation: 8x – 8y = 32 x – intercept: ( , ) y – intercept: ( , ) Slope:
Unit 2 – Solving Equations 26
Solving Standard Form for Slope Intercept Form
Examples:
Equation: 9x + 3y = 18
x – intercept: ( , )
y – intercept: ( , )
Slope:
Solve for Slope-Intercept Form: 9x + 3y = 18
Slope:
y – intercept: ( , )
x – intercept: ( , )
Equation: 2x + 8y = -8
x – intercept: ( , )
y – intercept: ( , )
Slope:
Solve for Slope-Intercept Form: 2x + 8y = -8
Slope:
y – intercept: ( , )
x – intercept: ( , )
Unit 2 – Solving Equations 27
Practice:
Equation: 5x – 10y = 30
x – intercept: ( , )
y – intercept: ( , )
Slope:
Solve for Slope-Intercept Form: 5x – 10y = 30
Slope:
y – intercept: ( , )
x – intercept: ( , )
Equation: 7x – 7y = 56
x – intercept: ( , )
y – intercept: ( , )
Slope:
Solve for Slope-Intercept Form: 7x – 7y = 56
Slope:
y – intercept: ( , )
x – intercept: ( , )
Continued…
Unit 2 – Solving Equations 28
Equation: 4x + 6y = -12
x – intercept: ( , )
y – intercept: ( , )
Slope:
Solve for Slope-Intercept Form: 4x + 6y = -12
Slope:
y – intercept: ( , )
x – intercept: ( , )
Equation: 10x – 5y = -40
x – intercept: ( , )
y – intercept: ( , )
Slope:
Solve for Slope-Intercept Form: 10x – 5y = -40
Slope:
y – intercept: ( , )
x – intercept: ( , )
Unit 2 – Solving Equations 29
Verifying Solutions
Solution (to a line): Any order pair that is a part of the line.
Is ( -1 , 4 ) a solution to
−2𝑦 + 2 = 6𝑥
Solve and graph
−2𝑦 + 2 = 6𝑥
Is ( 2 , -3 ) a solution to
−2𝑥 + 3𝑦 + 15 = 0
Solve and graph
−2𝑥 + 3𝑦 + 15 = 0
Is ( 4 , 2 ) a solution to
2𝑦 = 8
Solve and graph
2𝑦 = 8
Is ( -3 , 5 ) a solution to
𝑥 + 3 = 0
Solve and graph
𝑥 + 3 = 0
Unit 2 – Solving Equations 30
Practice:
Is ( 0 , -5 ) a solution to
−3𝑦 + 4𝑥 = 15
Solve and graph
−3𝑦 + 4𝑥 = 15
Is ( -5 , -2 ) a solution to
𝑥 + 𝑦 = 3
Solve and graph
𝑥 + 𝑦 = 3
Is ( 2 , 0 ) a solution to
3𝑥 + 2𝑦 − 8 = 0
Solve and graph
3𝑥 + 2𝑦 − 8 = 0
Is ( 2 , -2 ) a solution to
6𝑥 = −12
Solve and graph
6𝑥 = −12
Continued…
Unit 2 – Solving Equations 31
Is ( -6 , -1 ) a solution to
−2𝑥 + 6𝑦 = 6
Solve and graph
−2𝑥 + 6𝑦 = 6
Is ( 0 , -2 ) a solution to
4𝑦 − 3𝑥 = −8
Solve and graph
4𝑦 − 3𝑥 = −8
Is ( 2 , -2 ) a solution to
𝑦 − 5 = −3
Solve and graph
𝑦 − 5 = −3
Is ( 1 , 5 ) a solution to
−𝑥 + 𝑦 = −4
Solve and graph
−𝑥 + 𝑦 = −4
Unit 2 – Solving Equations 32
Unit 2 – Solving Equations 33
Unit 2 – Solving Equations 34