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Unit 2 – Solving Equations 1
Name: ____________________ Teacher: _____________ Per: ___
Unit 1
Unit 2
Unit 3
Unit 4
Unit 5
Unit 6
Unit 7
Unit 8
Unit 9
Unit 10
– Unit 2 – [Solving Equations]
3x – 2 = 4
3x – 2 = 4 + 2 +2
3x = 6
3x = 6 3 3
x = 2
Add 2 to both sides Divide both sides by 3
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 Sep 15 16 17 18 19
Classifying Numbers Number Properties 1-Step Equations
Evaluating Expressions
Combining Like Terms
Distributing
Solving 2 Step Equations
22 23 24 25 26
QUIZ Solving 2-Step Equations with Combining or
Distributing first.
Solving Multi Step Equations
Writing Equations and Solving
Review
TEST
29 30 Oct 1 2 3
Solving Equations with Variables on
Both Sides
Equations Review
Review 6-Weeks
TEST
Early Release Day
Essential Questions
What is the best method to solve a problem?
Why is it useful to write, transform, and/or solve equations that model situations?
What is a solution and how do I show that it is reasonable?
Unit 2 – Solving Equations 4
Critical Vocabulary
Real Numbers
Solution
Associative property
Commutative property
Distributive property
Algebra Tiles:
x2 x 1 -x2 -x -1
Unit 2 – Solving Equations 5
Classifying Numbers
Rational Numbers: Any number that ________ be written as a fraction / ratio. Irrational Numbers: Any number that ________ be written as a fraction / ratio. Whole Numbers: Any number that can be written without a __________ or a __________/__________ part.
Integers: Any number that can be written without a decimal/fractional part, but can be __________ or ___________.
Let’s place the following numbers: 4, -9, 0, ½, 5.3̅, π, 1.25, √9, √3
Rational Irrational
Integer
Whole
Unit 2 – Solving Equations 6
Number Properties
Unit 2 – Solving Equations 7
1-Step Equations
Equation Tiles Algebra Notation
x + 4 = 5
x + 4 = 5
x – 2 = 6
x – 2 = 6
4x = 8
4x = 8
𝑥
3 = 2
𝑥
3 = 2
Unit 2 – Solving Equations 8
Evaluating Expressions
For the following expressions: a = 2 b = -1 c = 5 x = -3 y = 6 z = -4
Examples:
ab – cy
2z2 + x a(xz – b)+2 z
Find the volume of a rectangular prism, V = lwh, if the length is 3, the width is 5, and the height is 12
Practice:
xy + z 3(b – x) abc + xyz
cx + za – 9
z(-4 + b) 2
√(𝑥2 + 𝑧2)
Find the perimeter of a rectangular, P = 2(l + w) if the width is 8 and the length is 9.
Find the density of an object, D = 𝑚
𝑣, if the volume is 8 and the mass is 32.
Kinetic Energy is found with the formula ½(mv2). Find the Kinetic Energy for an object with mass = 4 and velocity = 5.
Unit 2 – Solving Equations 9
Unit 2 – Solving Equations 10
Translating Words to Algebra
Sum
Combined Increased by
Plus ***Added to
***More than
Difference Decreased by
Minus Subtracted from
***Less than ***Fewer than
Times
Product Of
Twice Double/Triple
Is Are
Will be
Divided by Quotient
Per Half
Ratio
Plus the sum of … Twice the difference of … Minus the product of …
Unit 2 – Solving Equations 11
Examples:
The product of a number and 5
5x
Twice the difference of a number and 2
2(x-2)
3 more than twice a number
2x + 3
5 times the sum of a number and 3
5(x + 3)
The difference of a number and 4
x – 4
6 less than a number
x – 6
1 more than a number
x + 1
2 less than one-third of a number
1/3x – 2
3 subtracted from 4 times a number
4x – 3
Practice:
7 less than a number
Two-thirds of a number
Three times the sum of a number and 1
The quotient of a number and 10
The difference of a number and 7
3 more than a number
Twice the difference of a number and 5
9 more than twice a number
30 less than one-half of a number
17 subtracted from 4 times a number
2 more than triple a number
8 less than one-fifth of a number
Unit 2 – Solving Equations 12
Combining Like Terms
Must be the same ____________ and _____________ in order to combine.
The standard is to write the combined answer starting with largest exponents first and working down.
Examples With Algebra Tiles:
−7 + (−4)
−2𝑥 + 7𝑥 + 1
𝑥2 + 4𝑥 − 3𝑥
𝑥2 − 3𝑥 + 4 + 3x – 2
Examples Without Algebra Tiles:
−4𝑥 + 10𝑥
12𝑟 + 5 + 3𝑟 − 5
𝑥2 + 4 + 2𝑦2 − 6 + 5𝑥 − 3𝑦 + 𝑥2 − 𝑦2 − 2
7𝑥2𝑦 + 3𝑥2 − 2𝑦2 + 7𝑥2 − 5𝑥2𝑦
Unit 2 – Solving Equations 13
Application Problems:
Bob had three functions, g(x) = 2𝑥2 + 5𝑥 − 3, p(x) = 4𝑥 − 7, and s(x) = 3𝑥3 + 10. Find g(x) + p(x) + s(x).
Write an expression for the perimeter of the figure below:
Practice:
−10 + 3 + (−2)
5𝑥 + 3𝑥 + 7 − 𝑥 − 4
2𝑥2 + 5𝑥 + 6 − 3𝑥2
𝑥2 + 5𝑥 − 3 − 6𝑥 − 2𝑥2
Unit 2 – Solving Equations 14
Write an expression for the perimeter of the rectangle below:
10𝑎2 + 𝑏 + 5 − 2𝑎2
−7 + 5𝑥 − 3𝑥 + 𝑥2
−9𝑥 + 8𝑦 + 7𝑥𝑦 − 4 + 7𝑦 − 2𝑦
4𝑥 − 3𝑦 − 9𝑥 + 6𝑦 − 1
6𝑎 + 𝑏2 + 4𝑎2 − 8 − 2𝑎
𝑥2 + 5𝑥 − 5 − 9𝑥 − 6 + 4𝑦2
8𝑥2𝑦 − 𝑦2 + 4𝑥 − 6 + 3𝑦2 − 5𝑥2𝑦
7𝑥 − 40𝑦 − 6 + 4𝑥2 + 7𝑦 + 4
Unit 2 – Solving Equations 15
Distributing
Examples:
2(x – 4)
3(3 – x)
-3(x – 5)
(6 + 3y) 5
−3
4(36𝑥 + 60)
2(4x – 6) for x = -3
Unit 2 – Solving Equations 16
Practice:
6(4 – a)
5(n + 6) (3x – 5) 4
-4(3r − 8)
-2(e + 7)
−5(10x + 1)
- )( y25305
1
-(-2x + 5) 2
3(9𝑥 + 15)
−6(7 + x)
4(5z + 6) for z = -1 3(5 + 5x) for x = 2
Unit 2 – Solving Equations 17
Unit 2 – Solving Equations 18
Solving 2 Step Equations
Examples:
5x – 5 = -30
-x + 8 = 0
4x + 2y = 16 Solve for y
Unit 2 – Solving Equations 19
Practice: 6x + 1 = 37
- 10 + 3x = 14
-x + 4 = 9
- 6x - 9 = - 75
3x – 9 = 0
5x – 4 = -9
-4x + 9 = -15
-7x – 3 = -38
-21 = 8x + 11
𝑥
7+ 5 = 9
-12x + 3y = -6 Solve for y
- 4x + 4 = - 8
Unit 2 – Solving Equations 20
Solving 2 Step Equations with Distributing or Combining
Examples: -2(5x – 4) = -78
- 4x + 8 + 7x = 8
-75 = 5(-2x + 1)
x - 1 - 2x = - 5
2(l + w) = P Solve for w
Unit 2 – Solving Equations 21
Practice: 4(3x – 5) = -68
-10 = 7x + 2 – 9x
-2(x + 5) = -14
-4x + 3x – 8 = -7
6(2x + 1) = 18
45 = 5x + 5x - 5
-3(4x – 2) = 42
4x - 9 + 6x = 101
For a rectangle with a length equal to 3 more than its width, P = w + w + w + 3 + w + 3. Solve for w
Unit 2 – Solving Equations 22
Solving Multi Step Equations
Examples: 4)3(2 xx
9)1(315 x
x
−12=
5
6
8
h − 2=
4
3
V = lwh Solve for h
D = m
v Solve for v
Unit 2 – Solving Equations 23
Practice: d + 4(d + 6) = −1
1
2(12x − 10) = 31
5m − 2(m − 5) = 28
−10 + 4(3p + 10) = 18
8
b + 5=
10
5
6
𝑟 − 9=
3
2
33 = 3 + 5(y − 2)
x
−9=
2
3
I = prt Solve for r
5x+y
𝑎= 2 Solve for a
Unit 2 – Solving Equations 24
Writing Equations
Examples: Oceanside Bike Rental Shop charges twelve dollars plus eight dollars an hour for renting a bike. Alyssa paid eighty - four dollars to rent a bike. How many hours did she pay to have the bike checked out?
Sandy sold half of her comic books and then bought 8 more. She now has 11. How many did she begin with?
Benny bought five new baseball trading cards to add to his collection. The next day his dog ate half of his collection. There are now only thirty - seven cards left. How many cards did Benny start with?
Suppose it takes 48 chicken fingers to feed Mr. Young’s 4th grade class of 20 students. How many chicken fingers would be needed for 30 students?
Unit 2 – Solving Equations 25
Practice: Nancy bought a soft drink for 2 dollars and 8 candy bars. She spent a total of 26 dollars. How much did each candy bar cost?
Benny spent half of his allowance going to the movies. He washed the family car and earned 7 dollars. What is his weekly allowance if he ended with 13 dollars?
Danielle wants to purchase some clothes. She has $75. She buys one leather jacket for $39 and 3 pair of socks. Each pair of socks cost the same price. Determine the price of one pair of socks.
When ringing up a customer, a cashier needs 11 seconds to process payment as well as one second to scan each item being purchased. How many items would someone have to purchase in order for the transaction to take 37 seconds?
On Monday, 357 students went on a trip to the zoo. All 7 buses were filled and 7 students had to travel in cars. How many students were in each bus?
David read 40 pages of a book in 50 minutes. How many pages should he be able to read in 80 minutes?
Unit 2 – Solving Equations 26
Write an Expression for the Perimeter of the rectangle below:
Expression without Simplifying: _________________________ Simplify the expression by Combining Like Terms:___________
If the actual perimeter is known to be 42, write and solve the Equation for x:
Knowing x, what is the width? _______ What is the length? _______
Knowing both the width and length, what is the area? _________
Write an Expression for the Area of the rectangle below:
Expression without Simplifying: ____________ Simplify the expression by distributing: _____________
If the actual area is known to be 66, write and solve the Equation for x:
We know the width is 3, what is the length? _______
Knowing both the width and length, what is the perimeter? _________
Unit 2 – Solving Equations 27
Unit 2 – Solving Equations 28
Solving Equations with Variables on Both Sides
Examples: 3a +1 = a + 9
14 – 12x = 8x – 46
k + 5 = 20 – 2k
-5 + 7w = 8w + 12
Unit 2 – Solving Equations 29
Practice: pp 2022814
7981 zz
ww 1671239
vv 8234015
3263 xx
nn 81463
3538 uu
yy 4337
36244 xx
288 kk
Unit 2 – Solving Equations 30
Special Cases
When x = 0
Variable DOES NOT CANCEL, but the equation = 0
“x = 0”
2(3x + 4) – x = 8
No Solution
Variable CANCELS,
False Statement
“No Solution” or “NS”
2(3x + 4) – x = 5x + 1
Infinite Solutions
Variable CANCELS,
True Statement
“Infinite Solutions” or “All Real Numbers” or
“ℝ”
2(3x + 4) – x = 5x + 8
Unit 2 – Solving Equations 31
Unit 2 – Solving Equations 32
Equations Review / Extra Practice
2 Step Equations:
5x + 6 = 36
- 2x - 3 = - 11
- 3x + 3 = - 3
7x - 8 = - 29
2 Step Equations DISTRIBUTE FIRST:
6(5 - 3x) = - 114
- 2( - 6 + 7x) = 138
- 6( - 4x + 9) = 66
4( - 3x + 5) = - 100
Unit 2 – Solving Equations 33
2 Step Equations COMBINE LIKE TERMS FIRST:
5x + 8 + 2x = - 69
3 = - 2x + 6x + 7
- 5x - 4 + x = - 36
19 = 4x + 7 – 3x
2 Step Equations VARIABLES ON BOTH SIDES:
436 mm aa 913
9335 xx xx 10482
Unit 2 – Solving Equations 34