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Prayer
The Lord is my Shepherd, I shall not want. He makes me lie down in green pastures;
He leads me beside still waters;
He restores my soul. He leads me in right paths For His Name's sake.
Even though I walk through the darkest valley, I fear no evil;
For you are with me;
Your rod and Your staff - they comfort me.
You prepare a table before me in the presence of my enemies, You anoint my head
with oil;
My cup overflows.
Surely goodness and mercy shall follow me All the days of my life. And I shall dwell
in the house of the Lord My whole life long.
Solving Equations
Algebra I Unit 2
Warm Up for 8/22
Solving Equations
• When solving equations with variables in them, remember that we
should treat the equal sign like a scale
• Both sides always need to remain in balance
• In order to keep them balanced, we need to make us of the Identity
and Inverse Properties
Properties of Real Numbers
• Identity Property: Adding zero or multiplying by one does not change
the value of a term
• Inverse Property: Adding the opposite or multiplying by a reciprocal
makes the value of the expression either 0 or 1
Solving Equations
• We must also use the Addition and Subtraction Properties of Equality
Solving Equations
• To solve an equation with a variable, we must first Isolate the variable
• This means making sure the variable is the only term on one side of the
equation
• We can accomplish this by using inverse operations and the Addition
and Subtraction Properties of Equality
Solving Equations
Practice
• Solve the following equations for the given variable. Check Answers.
8 + x = 13
-4 + y = 5
g–6=3
r + 21 = -4
-13 + n = 4
t + (-5) = -10
Solving Equations
• When variables have a coefficient, we can use the Multiplication and
Division Properties of Equality to isolate the variables
Solving Equations
Practice
• Solve the following equations for the given variable. Check Answers.
6y = 54
𝑓
3
= 18
𝑡
8
=2
2.5r = 10
5h = 125
𝑥
4
= 13
Exit Card
Warm Up for 8/23
• Socrative.com
• Student Log-in
• Room Name: SIEVERSROOM
• Type your name
• Begin activity when logged in
Solving Equations
Solving Equations
Solving Equations
• Solve each equation for the given variable.
Solving Two-Step Equations
• When there are multiple terms on one side of an equation, you must
work backwards to isolate the variable
• Usually you do this by working in reverse Order of Operations
• Inverse Addition and Subtraction first
• Then inverse Multiplication and Division
Solving Equations
Solving Equations
Solving Equations
• Solve each equation. Show each step.
Solving Equations
Solving Equations
Solving Equations
• Solve each equation. Show each step.
Warm Up for 8/24
Exit Card
Warm Up for 8/25
• Simplify each expression by combining like terms.
1. −6𝑘 + 9𝑘
2. −4𝑥 − 10𝑥
3. 12𝑟 − 8 − 12
4. −2𝑥 + 11 + 6𝑥
5. −3𝑦 − 9 − 15𝑦
6. 𝑛 − 4 + 9 − 3𝑛
7. 12𝑟 + 5 − 3𝑟 − 5
8. −4 + 7(1 − 3𝑥)
9. −2𝑛 − (9 − 10𝑛)
Solving Equations
• We are able to easily solve One and Two-step equations
• Therefore, whenever we have equations that do not look quite like
the ones we have been working with, we need to get them to look
like a One or Two-step equation first
• We need to first get the side of the equation with the variables as
simplified as possible
Solving Equations
Solving Equations
Solving Equations
Solving Equations
Exit Card
Prayer
• Come Holy Spirit, fill the hearts of your faithful and kindle in them the
fire of your love.
• Send forth your Spirit and they shall be created. And You shall renew
the face of the Earth.
• O, God, who by the light of the Holy Spirit, did instruct the hearts of
the faithful, grant that by the same Holy Spirit we may be truly wise
and ever enjoy His consolations,
• Through Christ Our Lord
• Amen.
Warm Up for 8/29
Equations with Variables on Both Sides
• When equations have variables on both sides, we have to remember
to keep the equations balanced
• We need to eliminate the variables from one side and constants on
the other
• It is different from combining like terms, because we have to clear
them away
Equations with Variables on Both Sides
Equations with Variables on Both Sides
Practice
• Solve the following equations
1.) 8𝑥 − 8 = 32 + 3𝑥
2.) 3𝑥 + 4 = 6𝑥 − 11
3.) 4 𝑥 + 6 = 10𝑥
Practice
1. 4y + 15 = 6y – 11
2. 5p + 6 = –4p – 8
3. 6q – 1 = –q + 20
Exit Card
• Solve the equations
1. 13k + 5 = k – 7
2. –2m + 13 = 2m – 3
Warm Up for 8/30
• Solve each equation.
1. 13k + 5 = k – 7
2. 4(h + 2) = 3(h – 2)
3. 5x + 7 + 3x = –8 + 3x
4. 6(4z + 2) = 3(8z + 4)
Identities and No Solutions
• Sometimes equations can have no solutions. Other times, they can
have an infinite number of solutions.
• An equation has no solutions when you can simplify it down to a false
statement
• Example: 3𝑥 − 4 = 3(𝑥 + 2)
Identities and No Solutions
• When an equation has an infinite number of solutions, we call it an
Identity
• An equation is an Identity if you are able to simplify both sides of the
equation down to the same exact numbers on each side.
• Example: 5𝑥 + 6 − 2𝑥 = 3𝑥 + 6
Identities and No Solutions
• Tell whether each equation is an identity or has no solution. Show all
steps.
1.) 4(3m + 4) = 2(6m + 8)
2.) 5x + 2x – 3 = –3x + 10x
Identities and No Solutions
• Tell whether each equation is an identity or has no solution.
–8t + 3t + 2 = –5t – 6
–(8m + 4) = 4m – 2(6m + 2)
Exit Card
• Solve each equation for the given variable. Show all steps.
1.) 3n + 2 = –2n – 8
2.) 4t + 9 = –8t – 13
3.) 7(h + 3) = 6(h – 3)
4.) 14 + 3n = 8n – 3(n – 4)
Warm Up for 8/31
Modeling With Equations
• Steps for solving problems using modeling
1. Determine what you want to find in the problem
2. Determine what you are given in the problem
3. Think about how the variables and numbers are related
4. Write the equation
5. Solve the equation
6. Check that the answer makes sense in context
Modeling With Two Step Equations
Chip earns a base salary of $500 per month as a salesman. In addition
to the salary, he earns $90 per product that he sells. If his goal is to
earn $5000 per month, how many products does he need to sell?
1. What do we want?
2. What do we know?
3. How are they related?
4. Write and solve the equation
5. Check that answer makes sense in context
Modeling With Two Step Equations
A pizza shop charges $9 for a large cheese pizza. Additional toppings
cost $1.25 per topping. Heather paid $15.25 for her large pizza. How
many toppings did she order?
1. What do we want?
2. What do we know?
3. How are they related?
4. Write and solve the equation
5. Check that answer makes sense in context
Modeling With Two Step Equations
A pizza shop charges $9 for a large cheese pizza. Additional toppings
cost $1.25 per topping. Heather paid $15.25 for her large pizza. How
many toppings did she order?
1. What do we want?
2. What do we know?
3. How are they related?
4. Write and solve the equation
5. Check that answer makes sense in context
Warm Up for 9/1
• Solve each equation. If it is an identity or no solution, state so.
1. 25h + 40 = –15h – 80
2. –(3b – 15) = 6(2b + 5)
3. 18 – 6a = 4a – 4(a + 3)
4. –5(x + 7) = –5x + 35
Exit Card
Modeling With Multi-Step Equations
General admission tickets to the fair cost $3.50 per person. Ride passes
cost an additional $5.50 per person. Parking costs $6 for the family. The
total costs for ride passes and parking was $51. How many people in
the family attended the fair?
1. What do we want?
2. What do we know?
3. How are they related?
4. Write and solve the equation
5. Check that answer makes sense in context
Modeling With Multi-Step Equations
• Angela ate at the same restaurant four times. Each time she ordered a
salad and left a $5 tip. She spent a total of $54. Write and solve an
equation to find the cost of each salad.
1. What do we want?
2. What do we know?
3. How are they related?
4. Write and solve the equation
5. Check that answer makes sense in context
Modeling With Multi-Step Equations
• Eli took the fleet of 8 vans for oil changes. All of the vans needed
windshield wipers which cost $24 per van. The total bill was $432.
Write an equation to find out what each oil change cost. Solve the
equation.
1. What do we want?
2. What do we know?
3. How are they related?
4. Write and solve the equation
5. Check that answer makes sense in context
Modeling With Variables on Both Sides
• Eli took the fleet of 8 vans for oil changes. All of the vans needed
windshield wipers which cost $24 per van. The total bill was $432.
Write an equation to find out what each oil change cost. Solve the e
1. What do we want?
2. What do we know?
3. How are they related?
4. Write and solve the equation
5. Check that answer makes sense in context
Modeling With Variables on Both Sides
• Eli took the fleet of 8 vans for oil changes. All of the vans needed
windshield wipers which cost $24 per van. The total bill was $432.
Write an equation to find out what each oil change cost. Solve the
equation.
1. What do we want?
2. What do we know?
3. How are they related?
4. Write and solve the equation
5. Check that answer makes sense in context
Exit Card
• Eli took the fleet of 8 vans for oil changes. All of the vans needed
windshield wipers which cost $24 per van. The total bill was $432.
Write an equation to find out what each oil change cost. Solve the e
Warm Up for 9/7
• You want to buy some pizza for some friends and yourself. Each pizza
costs $8 and you need to tip the driver $10. If you have $74 to spend,
how many pizzas can you get? Set up an equation and solve
• You now want to order some breadsticks as well. They cost $4 each.
Set up an equation including both pizza and breadsticks using the info
from above. What are some possible combinations you can purchase?
Pizza and Breadsticks
• Equation for just Pizza:
• Number of Pizzas you can order:
• Equation for Breadsticks and Pizza:
• Number of Breadsticks and Pizzas you can order:
Pizza and Breadsticks
• Equation for just Pizza:
• Number of Pizzas you can order:
• Equation for Breadsticks and Pizza:
• Number of Breadsticks and Pizzas you can order:
Literal Equations
• An Equation that contains two or more variables
• Examples:
• Can we solve a literal equation?
Rewriting Literal equations
• Same process as solving equations: Isolate the variable you desire
• Treat other variables as if they are just constant numbers
5𝑥 − 6 = 29
5𝑥 − 𝑦 = 29
• Once we Rewrite the equation, we can “solve” for one variable when
given the others
𝑥=
29+𝑦
5
Literal Equations
Solve each equation for m. Then find the value of m for each value of n.
Exit Card
• Rewrite each equation so it is in the form of x=
1.) 3x+8=32
2.) x-5y=30
3.) 3x+8y=18
Warm Up for 9/8
• Solve each equation for y and find y for each value of x.
1.) y + 5x = 6; x = –1, 0, 1
2.) –3y = 2x – 9; x = –3, 0, 3
Literal Equations
• The properties that allow us to rewrite equations apply to variables
and constants, so we can rewrite literal equations that contain only
variables
Literal Equations
Literal Equations
• Practice solving for x
Literal Equations
• Rewrite each equation for p
xp + y = z
a = b + cp
Formulas
• Literal equations that contain only variables are sometimes called
formulas
• Formulas state the relationships between variable quantities
• Examples:
Common Formulas
Formulas
Formulas
Exit Card