©2013 Judo Math Inc.

SOLVING EQUATIONS

©2013 Judo Math Inc.

7th grade

Algebra/Proportions Discipline: Blue Belt Training

Welcome to the Blue Belt – Multi-step Equations Now that you have mastered the building block of algebra, linear expressions, you are ready to become a full blown algebraist. We are moving onto equations. Throughout history, equations made mathematicians happy, sad, confused, and amazed all at the same time! When a mathematician (like you) works with a situation for a long time and begins to notice a pattern, an equation is usually formed that makes life much easier. There are many famous equations like:

In this blue belt packet, however, we will mainly be looking at equations that can be solved in

just a few steps (we don’t want you to go crazy like many mathematicians before you did!)

Good Luck Grasshopper!

Order of Mastery:All about equations 1. Converting between Rational Numbers (review 7EE3) 2. Writing equations from a given situation (7EE4a) 3. Solving one step equations (7EE4a) 4. Solving two step equations (7EE4a) 5. Multi step equations with distributive property (7EE4a)

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1. Converting between Rational Numbers

It is VERY important to be able to manipulate fractions, decimals, and percents quickly and

easily. Particularly when you are working with equations! You will often run into some major

problems if you cannot add a fraction to a decimal… and so you must practice converting! You

most definitely learned this in previous grades, so create a table below with some quick

explanations to help you remember how to easily move between these.

Fraction -> Decimal Decimal-> Fraction

Decimal-> Percent Percent-> Decimal

Fraction-> Percent Percent-> Fraction

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Now complete these table, showing equivalent fractions, decimals, and percents:

Solve the following equations for x (you will have to convert!)Please express your answer as both a

fraction and a decimal.

1. 1

2 + 0.6 = x

2. 3.75+ 1

3 = x

3. 0.5+ 1

8 = x

4. 2.25+ 1

4 = x

5. 3.75+ 1

3 = x

6. 3.1+ 2

5 =x

Fraction Decimal Percent

1

3

0.125

76%

12%

4

5

1.0

3

7

120%

5

8

3

7. Jake and Sam are arguing about numbers again. Jake says that 1

4 and

1

5 are so close that

there are NO numbers in between them. Sam says he is crazy and that there are an INFINITE

number of numbers between them! Who is correct? Prove your solution by giving lots of

examples!

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2. Writing equations from a given situation

Equations arise from situations where something is unknown. In order to practice the art of equation

writing, we are going to work backwards. I will give you the “equation” and you can write the story

problem. After we have mastered this art, we can take on the challenge of writing equations.

Write 3 different story problems where the solution is based on the relationship

Equation: money earned - money spent on this - money spent on that = money left

1.

2.

3.

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Equation: original price - discount percent x original price = discounted price

1.

2.

3.

Equation: money earned each month - expenses each month = money to use each

month

1.

2.

3.

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Equation: speed x time = distance

1.

2.

3.

Now we will move on to writing some equations of our own. When doing this, it is important to …

1 Identify the quantities (what do you know and what do you

need to know? Give each of these a variable

2 Identify the relationship between the variables (Are they being

added, subtracted, multiplied, divided, etc?)

Now try this…

Helen has 2 inches of hair cut off each time she goes to the hair salon. If h equals the length of hair

before she cuts it and c equals the length of hair after she cuts it, which equation would you use to find

the length of Helen's hair after she visit the hair salon? Explain your solution.

a. h = 2 - c c. c = h - 2

b. c = 2 - h d. h = c – 2

Write about each answer giving the quantities and the relationship. I have done (part a) for you.

a. Length of hair before it’s cut = 2 take away the length of her hair after it’s cut.

b. __________________________________________________________________

c. __________________________________________________________________

d. __________________________________________________________________

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Check out these situations and see if you can write an equation to model them.

These have one variable and one operation…

a. A number increased by nine is fifteen.

b. Twice a number is eighteen.

c. Four less than a number is twenty.

d. A number divided by six is eight.

These have one variable and two operations…

a. Twice a number, decreased by twenty-nine, is seven.

b. Thirty-two is twice a number increased by eight.

c. The quotient of fifty and five more than a number is ten.

d. Twelve is sixteen less than four times a number.

Now try it with words…

a. Ellie is x years old. In thirteen years she will be twenty-four years old.

b. Each piece of candy costs 25 cents. The price of h pieces of candy is $2.00.

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c. Dan made a withdrawal of d dollars from her savings account. Her old balance was $350, and

her new balance is $280.

d. A large pizza pie with 15 slices is shared among p students so that each student's share is 3

slices.

Now a little more complicated… write and solve the equation that models each of these situations.

1. The attendance at a baseball game was 400 people. Student tickets cost $2 and adult tickets

cost$3.Total ticket sales were $1050.How many tickets of each type were sold?

2. Jeanne has $17 in her piggy bank. How much money does she need to buy a game that costs $68?

3. Eric had $197 in his savings account before he was paid his weekly salary. His current savings balance

is $429. How much money does Eric earn each week?

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3. Solving one step equations

A “one step equation” is an equation that can be solved in one step (just like the name says!)

Circle the equations below that can be solved in one step:3x=6

4x-8=10

x/3=7

4-x=8

5x=10

1/2x+5 = 7

The important thing to remember when solving equations is BALANCE. Say it out loud right now:

BALANCE!

Balance means whatever you do to one side of the equal sign, you must do to the other side as well!

IN the balance here, you can see the equation is x+3=8. In order to solve for x, I need to subtract 3. If I subtract it from one side, I better subtract it from the other

side as well! x+3=8

-3 = -3 X=5

To solve a one-step equation, do the inverse of whatever operation is being done to the variable. Because it is an equation, what is done to one side of the equation must be done to the other side of the equation.

Solve an addition equation by subtraction.

Solve a subtraction equation by addition.

Solve a multiplication equation by division.

Solve a division equation by multiplication.

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Write and solve an equation for each situation below. You may be able to solve the problem without writing an equation, but see if you can practice writing one anyway… I promise this practice will pay off eventually!

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3. Write three story problems that could be solved using a one-step equation and give them to a friend to solve.

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4. Solving two step equations

One step Equation Two step equation

x + 2 = 12 5x + 2 = 12

The above table shows a one-step equation and a two step equation side by side. Compare the two equations by answering these questions…

a. What are the differences?

b. What are the operations involved?

c. How do you think you would go about solving a two step equation?

It takes two steps to solve an equation or inequality that has more than one operation: The problem is, ORDER MATTERS! You can just do any operation in any order. It’s like when you were in 5th and 6th and you did a million problems like this to help you practice “PEMDAS”

Remember this?!

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The same applies with solving two step equations, only you have to go in reverse…. Follow these steps…

1 Simplify using the inverse of addition or subtraction.

2 Simplify further by using the inverse of multiplication or division.

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5. Multi step equations with distributive property

Just as with solving one-step or two-step or any equation, one goal in solving an equation is to have only variables on one side of the equal sign and numbers on the other side of the equal sign. The other goal is to have the number in front of the variable equal to one. There are a lot of different operations that you MIGHT do first when solving a multi-step equation and it’s all going to depend on the problem… bummer huh. But you are becoming master mathematicians and I know you can do it! Maybe these examples can help you…

Combine like terms FIRST 4x +2x-x=7

Move variables to one side FIRST 4x+9=2x-3

Distributive property FIRST 2z-5(z+1)=3z+1

Inspect the multi-step equation that is solved below. On the line write a quick note about what was done to arrive at that step. (example: combine like terms, distribute, inverse operation of +3)

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Distribute, combine like terms, and use opposite operations to solve the multi-step equations below.

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1. Write an equation that takes 1 step to be solved and solve it:

2. Write an equation that takes 2 steps to be solved and solve it:

3. Write an equation that takes 3 steps to be solved and solve it:

4. Write an equation that takes 4 step to be solved and solve it:

5. Write an equation that takes 5 step to be solved and solve it:

6. Just for fun, write an equation that takes 10 steps to solve and ask a friend to solve it!

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A few equation challenges before you go…

STUDENT VOLUNTEERS: Two student volunteers are stuffing envelopes for a local food

drive. The mailing will be sent to 560 possible donors. Luis can stuff 160 envelopes per hour

and Mele can stuff 120 envelopes per hour.

a. Working alone, what fraction of the job can Luis complete in one hour? In t hours?

b. Working alone, what fraction of the job can Mele complete in t hours?

c. Write an expression for the fraction of the job that Luis and Mele can complete in t hours if they

work together.

d. To find how long it will take Luis and Mele to complete the work together, you can set the

expression you wrote in part c equal to 1 and solve for t. Explain why this will work.

e. How long will it take Luis and Mele to complete the job if they work together? Check your

solution.

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Consecutive Integers: Consecutive integers are integers that follow each other in order (for

example 5, 6, and 7) You want to find 3 consecutive integers whose sum is 84.

a. Why does the equation n+ (n+1)+(n+2)=84 model this situation?

b. Solve the equation in part a to find the three consecutive integers.

Extra challenge: The sum of three consecutive numbers is 123. The second number is 9 less than 2 times

the first number. The third number is 6 more than three times the first number. Find the three

numbers.