Solving Equations

CCSM: A-REI. 1EXPLAIN each step in SOLVING a

simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution.  CONSTRUCT a viable argument to justify a solution method.

1. How can I write verbal expressions and equations from algebraic forms?

2. How can I write algebraic expressions and equations from verbal forms?

3. How are equations created to describe the relationship between numbers and variables?

4. How do you solve linear equations and inequalities in one-variable

General little statements that DO NOT have an equals sign!

2x – 4 2m³ + 5m Seven less than half a number

squared Twice the difference of a

number and five

Any statement that DOES have an equals sign (or equivalent):

4n + 7 = 11 ½ b – 2 = 3 Twice a number increased by four

is ten The difference of a number and two

is five

It means you will take an expression like 3x – 5, and write it as: “the difference of three times a number and five” or maybe “ five less than three times a number”

If you are given the verbal sentence, you will write it as an algebra equation or expression:

“half of a number increased by three is twelve” would be written as : ½ x + 3 = 12

Addition: add, plus, increased, sum, more than

Subtraction: minus, less than, decreased, subtract, declines

•Multiplication: multiply, per, per person, each, each customer, installments, percent, product, of

•Division: divided by, split, quotient

How would you write 4n + 3 as a verbal expression?

(THINK-PAIR-SHARE)

Can you write the following in algebraic form?

“3 less than a number squared is the same as five”

1. Twice a number less than seven

2. Seven decreased by twice a number

3. Seven less than twice a number

4. Twice a number decreased by seven

Two more than 3 times a number squared is fourteen

Three m’s squared plus two equals fourteen

Three times a number squared increased by two is the same as fourteen

It cost $5 per hour to rent golf carts from E-Z-GO. You need to rent golf carts for 10 hours. Write an equation. How much will it cost you?

Pair Share!!!!

What are the variables involved in this relationship?

Which variable is dependent? Which variable is independent?

Choose letters to represent your variables

What variable is the question asking you to find?

(what are you looking for?) **HINT**

• This is the dependent variable and is always the first letter you write; then you write the = sign.

Make a rule! (Say in words how you can find the

quantity for the variable you just wrote in step 4)

ASK YOURSELF: What calculations do I need to use? (+, -, x, ÷)

Addition: add, plus, increased, sum, more than, in addition

Subtraction: minus, less than, decreased, subtract, declines

•Multiply: multiply, per, each, installments, percent, of, times

•Divide: divided by, split, quotient, per

Write an equation using your rule

Test it! ASK YOURSELF: Does your equation

make sense?

What are the variables? What are the variables?

Choose the letters. Choose the letters. What are you looking for? What are you

looking for? Make a rule. Make a rule. Write an equation. Write an equation. Does it make sense?

The number of hot dogs needed for the picnic is two for each student

How many hot dogs do we need for 150 students?

The amount of material needed to make the curtains is 4 square yards per window.

How much material do we need for 5 windows?

You bring 30 cupcakes for your class and there are s amount of students. Write an equation to figure out how many cupcakes each student (s) will get.

If there are 15 students in the class, how many cupcakes will each student get?

It cost a flat fee of $25 plus $5 per hour to rent golf carts from E-Z-GO. You need to rent golf carts for 10 hours. Write an equation. How much will it cost you?

9-12-’12

A. A number that changes constantly.B. A letter or symbol that stands for a number.

C. A number that varies as you move away from zero.

If I took one block away from the left side, what

would I have to do to keep the scale from tipping?

If I added 6 more apples to the right side, how could I

keep the scale from tipping?

for Solving Addition and Subtraction Equations

1. Look to see what side the variable

is on.

2. Decide what

operation is being shown.

3. Do the opposite of

that operation to both sides.

4. Write what

number the variable

stands for (ex: a = 2).

5. Check the solution by “plugging” the number back into the equation to see if the sides are equal.

m + 18 = 34 Look for variable

-18 -18 Do the opposite

m = 16 Write what m is

m + 18 = 3416 + 18 = 34 Plug in m

34 = 34 Solution is correct!

68 = y + 43 Look for variable

-43 -43 Do the opposite

25 = y Write what y is68 = y + 4368 = 25 + 43 Plug in y 68 = 68 Solution is correct!

t - 14 = 22 Look for variable

+ 14 +14 Do the opposite t = 36 Write what t is t - 14 = 2236 - 14 = 22 Plug in t 22 = 22 Solution is correct!

1. r + 5 = 182. m - 26 = 593. 102 = x - 154. a + 39 = 56

r = 13m = 85117 = x a = 17

To get the variable by itself, which number needs to be

moved?-5

To move the -5, you have to do the opposite operation. What operation will we use?

division

-5 -5 t = -12

-5(-12) = 60

1. Draw “the river” to separate the equation into 2 sides

2. Divide both sides by -53. Simplify4. Check your answer

6 6 2.5 = n

15 = 6(2.5)

1. Draw “the river”2. Divide both sides by 63. Simplify4. Check your answer

1. v = -1262. v = -433. v = 434. v = 126

Answer Now

You don’t like fractions? Let’s get rid of them!

“Clear the fraction” by multiplying both sides of the

equation by the denominator.

x

4 12

x

4 12

4 · · 4 x = -48

1. Draw “the river”2. Clear the fraction –

multiply both sides by 43. Simplify4. Check your answer

x

4 12

248

14

3 · = 18 ·

3 2x = 54 2 2 x = 27

1. Draw “the river”2. Clear the fraction –

multiply both sides by 33. Simplify4. Divide both sides by 25. Simplify6. Check your answer

2( )2718

3

2x

32x

3

1. Multiply by 32. Multiply by 53. Multiply by -124. Multiply by -5

312

5

b

Answer Now

1. b = -562. b = -143. b = 144. b = 56

48

7

b

Answer Now

Using a Verbal Model

JET PILOT A jet pilot is flying from Los Angeles, CA to Chicago, IL at a speed of 500 miles per hour. When the plane is 600 miles from Chicago, an air traffic controller tells the pilot that it will be 2 hours before the plane can get clearance to land. The pilot knows the speed of the jet must be greater then 322 miles per hour or the jet could stall. a. At what speed would the jet have to fly to arrive in Chicago in 2 hours?

Is it reasonable for the pilot to fly directly to Chicago at the reduced speed from part (a) or must the pilot take some other action?

b.

LABELS

VERBAL MODEL

Using a Verbal Model

Speed ofjet • Time = Distance to

travel

Speed of jet =x

Time = 2

Distance to travel =

600

(miles per hour)

(miles)

(hours)

600=

x = 300

ALGEBRAIC MODEL

a. At what speed would the jet have to fly to arrive in Chicago in 2 hours?

2 x

SOLUTION

To arrive in 2 hours, the pilot would have to slow the jet down to 300 miles per hour.

You can use the formula (rate)(time) = (distance) to write a verbal model.

Using a Verbal Model

It is not reasonable for the pilot to fly at 300 miles per hour, because the jet could stall. The pilot should take some other action, such as circling in a holding pattern, to use some of the time.

Is it reasonable for the pilot to fly directly to Chicago at 300 miles per hour or must the pilot take some other action?

b.

Writing an Algebraic Model

You and three friends are having a dim sum lunch at a Chinese restaurant that charges $2 per plate. You order lots of plates. The waiter gives you a bill for $25.20, which includes tax of $1.20. Use mental math to solve the equation for how many plates your group ordered.

Understand the problem situation before you begin. For example, notice that tax is added after the total cost of the dim sum plates is figured.

SOLUTION

LABELS

VERBAL MODEL

Writing an Algebraic Model

Cost perplate • Number of

plates= Bill Tax–

Cost per plate =2

Number of plates =p

Amount of bill =25.20

Tax =

1.20

(dollars)

(dollars)

(dollars)

(plates)

25.20 1.20–2 =p

2p = 24.00p = 12

Your group ordered 12 plates of food costing $24.00.

ALGEBRAIC MODEL