algebra 1 review by thomas siwula. addition property (of equality) multiplication property (of...

Algebra 1 Review

By Thomas Siwula

Addition Property (of Equality)

Multiplication Property (of Equality)multiply the same number to each side

Example a=b, then a+c = b+c

Example: a=b then ac=bc

Reflexive Property

Symmetric Property

Transitive Property

If one number equals itself

Example: a=a

The numbers flipped around still equal each other

Example: If a=b then b=a

If one number equals a number that equals a third number the first number is equal to the third number

Example: If a=b and c=b then a=c

Associative Property of Addition

Associative Property of Multiplication

If the parentheses switch the outcome is still the same

Example: (a+b) +c = a+(b+c)

Example:

(-3 x 7)x 5 = -3x (7 x 5)

Commutative Property of AdditionThe order of the addition is switched and the sum is still the same

Commutative Property of Multiplication

The order of the numbers being multiplied is switched

Example: a+b=b+a

Example: axb=bxa

Distributive Property (of Multiplication over Addition

Multiply the numbers in the parentheses by the number outside of the parentheses.

Example: 7(2+3)= 14+21

The seven was distributed to the 2 and 3

Prop of Opposites or Inverse Property of Addition

A number plus its opposite is equal to zero

Prop of Reciprocals or Inverse Prop. of Multiplication

Example: x =1

Example: 3+(-3)=0

2

1

1

2

Identity Property of Addition A number plus zero equals itself

Identity Property of MultiplicationA number times one equals itself

Example: 4+0=4

Example: 5x1=5

Multiplicative Property of Zero Any number times zero equals zero

Closure Property of AdditionIf two real numbers are added together their sum will be a real number

Closure Property of Multiplication

Example: 13x0=0

Example: The real numbers 10+9=19, another real number

Example: The real numbers 4x6=24, another real

number

Product of Powers Property

Power of a Product Property

Power of a Power Property

Example:n3xn4=n7

Example: (RS)11=R11S11

Example: (p7)3=p21

Quotient of Powers Property

Power of a Quotient Property

Example: w9/w6=w3

Example: (7/3)2= 49

9

Zero Power PropertyAnything to the power of zero is 1

Negative Power PropertyAnything put to the negative power is put under 1 to

make positive

Example: (13)0=1

Example:7-2= 2 7

1

Zero Product Property If one variable in the equation equals zero

the equation will equal zero

Example: (n-10) (n-7)=0, therefore n-10 equals 0 or n-7 equals zero

Product of Roots PropertyExample:

√28=√4√7 Quotient of Roots Property

Example:√64/√4=√16=4

Power of a Root PropertyExample:

52=25

QuizLook at the sample problem and give the name of the property illustrated.

1. a=b then ac=bc

Click when you’re ready to see the answer.

Answer: Multiplication Power of Equality

QuizLook at the sample problem and give the name of the property illustrated.

1. 4+0=4

Click when you’re ready to see the answer.

Answer:

Identity Property of Addition

QuizLook at the sample problem and give the name of the property illustrated.

1. 3+(-3)=0

Click when you’re ready to see the answer.

Answer:

Prop of Opposites or Inverse Property of Addition

QuizLook at the sample problem and give the name of the property illustrated.

1. 13x0=0

Click when you’re ready to see the answer.

Answer:

Multiplicative Property of Zero

QuizLook at the sample problem and give the name of the property illustrated.

1. 7(2+3)=14+21

Click when you’re ready to see the answer.

Answer:

Distributive Property (of Multiplication over Addition

QuizLook at the sample problem and give the name of the property illustrated.

1. The real numbers 10+9=19, which is another real number

Click when you’re ready to see the answer.

Answer:

Closure Property of Addition

QuizLook at the sample problem and give the name of the property illustrated.

1. If a=b, then a+c = b+c

Click when you’re ready to see the answer.

Answer:

Addition Property of Equality

QuizLook at the sample problem and give the name of the property illustrated.

1. x =1

Click when you’re ready to see the answer.

Answer: Prop of Reciprocals or Inverse

Prop. of Multiplication

2

11

2

QuizLook at the sample problem and give the name of the property illustrated.

1. 3+5=5+3

Click when you’re ready to see the answer.

Answer:

Commutative Property of Addition

QuizLook at the sample problem and give the name of the property illustrated.

1. The real numbers 4x6=24, another real number

Click when you’re ready to see the answer.

Answer:

Closure Property of Multiplication

QuizLook at the sample problem and give the name of the property illustrated.

1. a=a

Click when you’re ready to see the answer.

Answer:

Reflexive Property

QuizLook at the sample problem and give the name of the property illustrated.

1. a=a

Click when you’re ready to see the answer.

Answer: Product of Roots Property

QuizLook at the sample problem and give the name of the property illustrated.

1. (a+b) +c = a+(b+c)

Click when you’re ready to see the answer.

Answer:

Associative Property of Addition

QuizLook at the sample problem and give the name of the property illustrated.

1. (-3 x 7)x 5 = -3x (7 x 5)Click when you’re ready to see the answer.

Answer:

Associative Property of Multiplication

QuizLook at the sample problem and give the name of the property illustrated.

1. AxB=BxAClick when you’re ready to see the answer.

Answer:

Commutative Property of Multiplication

QuizLook at the sample problem and give the name of the property illustrated.

1. a=b b=a

Click when you’re ready to see the answer.

Answer: Symmetric Property

1st Power Equations

• These equations have one variable in them to the first power. Solve by adding, subtracting, multiplying or dividing each side to get an answer.

Ex: x+11=5, subtract eleven from each side to get x by itself and then x = -6 is the answer.

Fractions with the 1st power

Ex: , find the LCD, which in this case is 21. Multiply that to all parts of the equation to cancel into-

Therefore 10x=210 and x= 21

-Equations with variable in the denominator-

Ex: ,find the LCD, which is 20a and multiply to all sides to cancel into 10-12a=24, subtract 10 from each side to get,-12a=12, divide and a=-1

107 3

x x

3 7 210x x

2 324

4 5a

Solving 1st Power Inequalities in One Variable A. With one inequality sign- 5<X

B. Conjunction- -3< X< 7the word “and” can also be used in conjunctionsC. Disjunction - 4>X or 2-3<X

D. Special Cases 1. x<-3 and x>7 = because x cannot be greater than seven and less than -3 at the same time

Linear Equations in Two Variables SlopesA. Positive slope -Rises from bottom left to top right. Ex: y=3/2x+7, this has

positive slope because 3/2 is positive. B. Negative slope-Otherwise known as falling lines and normally start at top left falls to bottom right. Ex: y= -5x + 2, this has a negative slope because there is a negative C. Vertical slope- Occurs when y equals zero and x equals a number Ex: x=3 The line will run vertically up and down the graph with a slope that is undefined.D. Horizontal slope- Occurs when x equals zero and y equals a number Ex: y=9 The line runs horizontally across the graph and the slope equals zero.-GraphingIn an equation such as y=3/2x+7, 7 is the y intercept so that would be plotted on the y axis on the graph. From the point 7, since the slope is 3/2 one would count up three and over two to graph the linear equation. The final product would look like this.

Linear Equations Continued

• There are two different types of slope form

1. Standard Form- Ax + By=C

2. Slope Intercept Form- Y=mx+b

Finding Slope and Intercept Points

Slope Formula- Gets the slope of the equation

Point Slope Formula- Once the slope is found, this formula finds the y intercept, if it is unknown.

To find out the x intercept make y equal to zero

To find out the y intercept make x equal zero or use point slope formula

Linear SystemsA. Substitution Method

Substitute an equation for a variable.Ex: 9x+y=4, when y is isolated the equation is y=5x+4 -5x+3y=2, substitute 9x+4 for y, so the equation turns into - 5x+3(5x+4)=2. This then is equivalent to -5x+15x+12=2, simplify and it is 10x=10, where 1 is equal to x. Plug 1 into the first equation and y equals 9. The answer is then (1,9)

B. Elimination Method

Eliminate one variable multiplying them by the LCF.Ex: -3y-7x=6 Times 2x by 7 and -7x by 2 so they cancel eachother out

7y+2x=10-3y-14x=67y+14x=10-3y=67y=104y=16, therefore y=4 and then plug that into one of the original problems.

7(4)=2x=10, which simplified is 28+2x=10, 18=2x and x equals 9The solution is (9,4)

Linear Systems Continued

• After solving the linear equations and graphing them, the lines will either be

1. Dependent-the equations both have the same exact line. Dependent is also a consistent line.

2. Consistent- There will be one point of intersection between the two lines

3. Inconsistent-The lines are parallel and will never intersect.

Ways to Factor

1. PST2. GCF3.Difference of Squares4. Sum and Difference of Cubes5.Reverse Foil6.Grouping 2 by 27. Grouping 3 by 1

Rational expressions A. Simplify by factor and cancel Factor the equation into conjugates and then cancel the common factors. This leaves the equation in its simplest

form Ex:

B. Addition and subtraction of rational expressions, factor, find the LCD, and multiply, add the numerators and cancel all common factors 

2 2 8 ( 2)( 4) 2

2 9 20 ( 4)( 5) 5

x

x

x x x x

x x x x

Multiplication and division of rational expressions

• For multiplication factor and cross cancel

• For division equations, flip the numerator and denominator around to multiply

Ex:  3x2 - 4x              x(3x - 4)             3x - 4   ------------      =    --------------      = ---------- 

2x2 - x                x(2x - 1)             2x - 1

Quadratic Equations in One Variable

• Factoring

- Set the equation to zero, then factor.

Ex: 8x2- 40x=0

8x (x-5) = 0 8x = 0 and (x-5)=0Solution x = (0, 5)

Ex: x2=36, set to zero, x2-36=0

(x-6) (x+6)= 0, therefore x = (-6,6)

Quadratics Continued

• Square Root of Both Sides -Take the square root of the variable and the numberEx: x4=25, take the squares of each side , 25 must have a plus minus sign in front of

it because x could equal + or – 5 x2=Ex: x2=24, square, Since 24 is not a perfect square you divide it into two

different squares, and then finish the problem is the final answer

4 25x

52 24x

4 6x

2 6x

Quadratics• Completing the Square

-Set a equal to 1 and ALWAYS put the equation into standard form

Ex: 4x2+24=32, set to standard form

4x2+24-32=0, set a equal to 1

x2+6-8=0, now add c to the other side

x2+6 =8, add the equation ( )2 to both sides

x2+6+9=8+9, this is a PST, so factor

(x-3)2=17, take square roots from each side

simplify into,

2

b

2( 3) 17x

3 17x

Quadratics

• Quadratic Formula-

Works with the same equation, 4x2+24=32

Put into standard form, 4x2+24-32=0

Plug into formula and solve

b2-4ac is called the discriminate.

It is used to find out if a certain equation has

-two irrational roots-number is a non positive square

-two rational roots-when number is positive square

-one rational double root-when number is equal to zero

-no real roots-when number is negative

2 4

2

b b ac

a

FunctionsA. In a function, f(x) stands for y.

B. Finding Domain and RangeDomain(x)- set y, also known as the range to zero, and then factor to find the domainRange(y)-set x also known as the domain to zeroC. When given two ordered pairs of data such as (5,-8), (4,7) to see these points on a graph 1. Use the slope formula to get the slope2. Use the point slope formula to find the y intercept and then graph the equation.

Quadratic functions• When graphing a quadratic function the

graph will be a parabola.

• If a is negative the parabola opens downwards, if a is positive the parabola opens upwards.

Find the x and y intercepts by the means that were stated in the previous slide.

Vertex equation- {-b/2(x)}

Axis of Symmetry- The axis = whatever the vertex is

Graphing A Parabola

• Ex: f(x)=x2+6x+9

X intercepts- 0=x2+6x+9

Factor (x+3) (x+3), x intercepts= (-3,0)

Y intercept- y=0-0+9, y intercept = (0,9)

Vertex (-6/2x1), -3 = vertex

- Plug -3 back into equation

f(-3)=9-18+9=0

Axis of Symmetry = 0

Simplifying expressions with exponentEx: 1. n10xn3=n13

2. 38/33= 35 or 33/38=1/35

3. (11n2p5)3= 33n6p15

4.-(72a3b2c-4)0=-1 5.x-3/5 - Flip the equation around to make exponent positive Answer is 5/x3

Simplifying expressions with radicals -Radical expressions are square roots that are not perfect so they need to be broken down before any squares can be found

Ex. √24=√4√6=2√6 Ex: √320=√64x5=4√5

Word Problems-1. A model airplane is propelled upward with a start speed of 36 ft/s. After how many seconds does it return to the ground? Plug the data into the equation h =rt-16t2 , where h is height, r is rate, and t is time. The starting equation will look like this- h=36t-16t2

Solve for t by means of GCF and factoring

2. In 3 days Jane lost 8 pounds, and then in 9 days Jane had lost 20 pounds. If the growth continues linearly, write an equation Jane could use to predict her weight on day 9. (Hint: Use the slope formula and the point slope formula to help with an answer.) To solve this plug in the data to the point slope and slope formula to make an equation that would solve the problem.3. A jar contains 19 coins in quarters and dimes, if the total value of the coins is 2.85, how many of each coin is there?To solve make variables for quarters and dimes. Then make the variables added together like this q+d=19. Plug the variables into this equation.

Word Problems• 3. A jar contains 19 coins in quarters and dimes, if the total value of the coins is 2.85,

how many of each coin is there?To solve make variables for quarters and dimes. Then make the variables added together like this, q+d=19 and isolate one variable, like so d=19-q. Plug the variables into this equation. 25q+10d=285 and then use the substitution method.

25q+10(19-q)=285. 25q+190-10q=285 Solve this equation solving for q and then plug q back into the equation of d=19-q to solve for d.

4.The length of a rectangle is 6 more than twice the width. The perimeter is 94. Find the dimensions of the triangle.

-Set the variable w for the width and formulate an equation.

-The equation would look like this 2(2w+6)+2w=96. The 2(2w+6) stands for the length of each side of the rectangle and the rest of the equation, 2w is for the width.

Solve- 4w+12+2w=96

6w+12+96

6w= 84 and w = 14

Line of Best Fit or Regression LineA. Line of Best Fit Regression is most commonly

used when predicting. Line of Best can predict the values of a dependent variable when compared with the values of an

independent variable.B. A calculator helps greatly with regression lines because one can simply plug in dependent and independent data into the calculator. The calculator will put that information into a graph and create a regression line that is ideal for making predictions of a certain variable.

Regression ProblemC. Try and find what the price of the house will be in year 7 using line of best fit and your calculator

Years House is For Sale

Price Decline

1 200,000

2 196,000

3 183,000

4 175,000

5 167,000