Solving for the Unknown; a How-

to approach to Solving Equations

Chapter 5

If no number is in front of a letter, it is

a 1.

B = 1B

C = 1C

If no sign is in front of a letter or

number, it is a + (positive value)

C = +C

4 = +4

Whatever you do to one side of an

equation, you must do to the other

Side (unless you are just

combining terms).

Solving for the Unknown Rule

Equation Equality Rule

You can add the same quantity or number to

both sides of the equation and subtract the

same quantity or number from both sides of

the equation without affecting the equality of

the equation.

You can also divide or multiply both sides of

the equation by the same quantity or number

(except 0) without affecting the equality of the

equation.

A + 8 = 58

Subtract 8 from Both Sides

A + 8 = 58

- 8 - 8

That leaves one A that equals 50

A + 8 = 58

- 8 - 8

A = 50

Opposite Process Rule

If an equation indicates a process

such as addition, subtraction,

multiplication, or division, solve for

the unknown or variable by using the

opposite process.

4Y = 28

4Y = 28

4 4

You can divide 4 into the 4 on the left of the

equation as well as the 28 on the right side of

the equation.

So…

4Y = 28

4 4

1Y = 7

or

Y = 7

Divide each side by

4 to find out how

much one Y is.

One Y is worth 7

Multiple Processes Rule

When solving for an unknown that

involves more than one process,

do the addition and subtraction

before the multiplication and division.

X + 5 = 50

9

X + 5 = 50

9 -5 -5

X = 45

9

X * 9 = 45(9)

9

X = 405

Subtract 5 from both sides

Original equation

Left after subtracting 5 from

both sides

Multiply each side by 9

After multiplying each side

by 9

When equations contain parentheses, (), which indicates

grouping together, you solve for the unknown by first,

multiplying each item inside the parentheses by the

number or letter just outside the parentheses.

Then you continue to solve for the unknown with the

opposite process used in the equation.

Do the addition and subtractions first, then the

multiplication and division.

Math Inside the Parentheses First

4(3F - 2) = 64

12F - 8 = 64

12F - 8 64

+ 8 = + 8

12F = 72

12F 72

12 12

F = 6

=

Multiply inside parentheses first

Add 8 to both sides

Divide both sides by 12

Solution-Unknown F equals 6

Result of multiplication

To solve equations with like

unknowns, you first combine

the unknowns and then solve

with the opposite process

used in the equation.

3Y-Y = 52

Like terms are the two Ys

3Y - Y = 52 original equation

2Y = 52 combining

unknowns (Y)

2Y = 52 divide by 2

2 2

Y = 26 solution

Solving Word Problems for Unknowns

1) Read the entire

problem.

2) Ask: “What is the

problem looking for?

3) Let a variable

represent the

unknown

4) Visualize the

relationship

between the

unknowns and

variables. Then set

up equation

5) Check your

results

C = Computers

4C + C = 600

Situation 1

Wal-mart reduced the price for a radio-alarm by $30.

The sale price is now $50. What was the old price?

P = Price

P–30 = 50

+30 +30

P = 80

Situation 2

Tribble spends ¼ of his allowance on toy

mice. He spends $2 on these mice. How

much is his allowance? A=allowance

¼ A = 2

4(¼ A) = 4( 2)

A = 8

Situation 3

John and Fumiko sell computers. John sells 3 times as

many computers as Fumiko. The difference is 20. How

many computers do they each sell? F = Fumiko

3F – F = 20

2F = 20

2F/2 = 20/2

F = 10

Fumiko sells 10 and John sells 30. He sells 3 times as

many as she does, and the difference is 20.

Situation 4

John sells 3 times as many computers as Fumiko. The

total number of computers sold is 40. F=Fumiko

F + 3F = 40

4F = 40

4F/4 = 40/4

F = 10

Fumiko’s sales + 3 times her sales = 40

Combining unknowns

Dividing both sides by the same number

Unknown is solved

Situation 5

•Lakers jerseys sell for $81 each and Clippers

jerseys for $6 each.

•Total sales were $498.

•People bought 3 times as many Lakers jerseys

as Clippers jerseys.

A. How many of each were sold?

B. What were the total dollar sales of each?

Solve for number of Clippers jerseys sold first, C.

# of Price

Jerseys per

Product Sold * Jersey = Total sales

Clippers jerseys C * $6 = $ 6C

Lakers jerseys 3C * $81 = $ 243C

Information Grid on Jersey Sales

The variable C = the number of

Clippers jerseys sold

$243C + $6C = $498 (total sales)

$249C = $498 (total sales)

$249C = $498

249 249

C = 2 [Clippers jerseys] $6 = $12 of sales)*

Combine the like terms

Finding out how

much 1 C is worth

Remember that three times more Lakers jerseys

were sold than Clippers jerseys.

3 times the 2 Clippers jerseys (C) = 6 Lakers jerseys

6 Lakers jerseys times $81 each = $486 of sales

Proof

6 Lakers jerseys * $81 = $486

+ 2 Clippers jerseys * $6 = + 12

$498