8th Grade Study Guide

System of Equations - Pythagorean Theorem - Laws of Exponents

Scientific Notation - Solving Equations

Solving a system of equations by Substitution substi

Step 1: Solve an equation for one variable.

Step 2: Substitute

Step 3: Solve the equation.

Step 4: Plug back in to find the other variable.

Step 5: Check your solution.

Pick the easier equation. The goal

is to get y= ; x=

Put the equation solved in Step 1

into the other equation.

Get the variable by itself.

Substitute the value of the variable

into the equation.

Substitute your ordered pair into

BOTH equations.

x + y = 5y = 3 + x

Step 1: Solve an equation for one variable.

Step 2: Substitute

The second equation is

already solved for y!

x + y = 5x + (3 + x) = 5

Step 3: Solve the equation.2x + 3 = 5

2x = 2

x = 1

x + y = 5y = 3 + x

Step 4: Plug back in to find the other variable.

x + y = 5

(1) + y = 5

y = 4

Step 5: Check your solution.

(1, 4)

(1) + (4) = 5

(4) = 3 + (1)

The solution is (1, 4).

System of Equations: Elimination

EXAMPLE #1:

STEP 2: To eliminate: Are they the SAME number and positive and negative in the Column? 5x + 3y =1 -1 (5x - 2y =1) use the distributive property

5x + 3y = 11

5x - 2y = 1

STEP 3: Solve for the variable. 5x + 3y =11

-5x + 2y = -1 5y =10 y = 2

STEP1: Draw your columns down the equation 5x + 3y =1 5x - 2y =1

STEP 4: Solve for the other variable by substituting

into either equation.5x + 3y =11

5x + 3(2) =11 5x + 6 =11 5x = 5 x = 1The solution to the system

is (1,2).

Pythagorean Theorem

Manipulated Formula:

C =

A =

B=

FOR EXAMPLE:The number 4,500 is written in scientific notation as ________________.4.5 x 103

The coefficient is _________.The coefficient must be a number greater than or equal to 1 and smaller than 10.

4.5

The power of 10 or exponent in this example is ______.

3

The exponent indicates how many times the coefficient must be multiplied by 10 to equal the original number of 4,500.

1. Move the decimal to the right of the first non-zero

number.2. Count how many places the

decimal had to be moved.3. If the decimal had to be moved

to the right, the exponent is negative.

4. If the decimal had to be moved to the left, the exponent is positive.

To write a number in scientific notation:

Addition and Subtraction

To add or subtract numbers written in scientific notation, you must….express them with the same power of ten.

Sample Problem: Add (5.8 x 103) and (2.16 x 104)

Solution: Since the two numbers are not expressed as the same power of ten, one of the numbers will have to be rewritten in the same power of ten as the other.

5.8 x 103 = .58 x 104 so .58 x 104 + 2.16 x 104 =?

Answer: 2.74 x 104

Multiplication

When multiplying numbers written in scientific notation…..multiply the first factors and add the exponents.

Sample Problem: Multiply (3.2 x 10-3) (2.1 x 105)

Solution: Multiply 3.2 x 2.1. Add the exponents -3 + 5

Answer: 6.7 x 102

Division

Divide the numerator by the denominator. Subtract the exponent in the denominator from the exponent in the numerator.

Sample Problem: Divide (6.4 x 106) by (1.7 x 102)

Solution: Divide 6.4 by 1.7. Subtract the exponents 6 - 2

Answer: 3.8 x 104

#2: Multiplying Powers: If you are multiplying Powers with the same base, KEEP the BASE & ADD the EXPONENTS!

m n m nx x x

So, I get it! When you multiply Powers, you add the exponents!

512

2222 93636

#3: Dividing Powers: When dividing Powers with the same base, KEEP the BASE & SUBTRACT the EXPONENTS!

mm n m n

n

xx x x

x

So, I get it!

When you divide Powers, you subtract the exponents!

16

222

2 4262

6

#4: Power of a Power: If you are raising a Power to an exponent, you multiply the exponents!

nm mnx x

So, when I take a Power to a power, I multiply the exponents

52323 55)5(

#5: Product Law of Exponents: If the product of the bases is powered by the same exponent, then the result is a multiplication of individual factors of the product, each powered by the given exponent.

n n nxy x y So, when I take a Power of a Product, I apply the exponent to all factors of the product.

222)( baab

#6: Quotient Law of Exponents: If the quotient of the bases is powered by the same exponent, then the result is both numerator and denominator , each powered by the given exponent.

n n

n

x x

y y

So, when I take a Power of a Quotient, I apply the exponent to all parts of the quotient.

81

16

3

2

3

24

44

Solving Equations

Left Column Right Column

1. Do you need to apply the rules for integers?

2. Do you need to do the distributive property?

3. Do you need to combine like terms?

4. Is the variable & coefficient next to the equal sign?

5. Did you isolate the variable?

6. Did you inverse the operation?

1. Is the number by itself?

2. Do you need to inverse the variable?

Did you draw a line down the equal sign?

Solve it!!