Algebra I SOL Topics and Formulas

Equations, Inequalities, Expressions

One Step Equations Addition/Subtraction:

(Perform the inverse “opposite” operations on each side of the equations) -3 -3

x = 7 **When you see addition, you subtract the value on both sides of the equal sign.

** When you see subtraction, you add the value on both sides of the equal sign. +3 +3 x = -7 One Step Equations Multiplication/Division:

**When you see a number next to the variable, it indicates multiplication. Therefore, you are going to divide each side by the leading coefficient (the number in front of the variable)

x = -5

(4) (4) ** When you see a number in the denominator that indicates division. Therefore, you are going to

multiply each side by the value in the denominator. X = -12

Multi-Step Equations:

1. Add or subtract using the inverse operations on both sides of the equal sign.2. If you see a number in the denominator then multiple both sides of the equal sign by the value in the

denominator.3. If you see a number next to a variable, divide each side of the equal sign by the coefficient.

**Notice that you may not need to perform every step in every problem.

(5) (5)

+4 + 4 2x = 19

In calculatorGo to MenuEquationSolverType in the equation exactly how you see it ( include ( ) if needed)Hit exe TWICE!!

Inequalities:Use the same steps as solving equations. The only difference if when you divide or multiply by a negative, you switch the symbols

Verbal ExpressionsOrder of Operations PEMDASTwo less than a number x-2

Remember these key words: ADD SUBTRACT MULTIPLYmore than less than (switch the numbers) productincreased by how much more? twicetotal of difference of (like half of)sum how much farther?

at least → ≥DIVIDE is – equals no more than → ≤shared equally is less than → < at most → ≤quotient is greater than → > no less than → ≥

“times the sum of” or “times the difference of” – you need parentheses

So “Five less than three times the sum of a number and seven” means 3(n+7) - 5

Example: Omar wants to have a party. He will pay $300 for a DJ and $10 per person for food. If his bill was $650, how many people are coming to the party?

Omar has to pay $300 for the DJ no matter what, that is the thing that doesn’t change. His cost for food changes according to how many people he invites. $650 is his total, so this is the equation:

300 + 10x = 650

You can use solver to get x = 35 people.

SOL 3: Properties

Distributive Property: Multiply a number outside of a parenthesis to every term inside.

6x – 12 = 6(x – 2)3(n – 7) = 3(n) + 3(-7)a(x + b) = ax + ab

Identity Property of Multiplication – if you multiply any number by 1 you get the original number

Identity Property of Addition – if you add zero to any number you get the original number

We divided by a negative….flip the sign!

Commutative Property :the numbers can change places (or order)3 + 5 = 5 + 3ab = ba(ab) + 3 = (ba) + 3(7a + 9 ) +5a = 5a + (9 + 7a)

There is no commutative property of division or subtraction

Reflexive - the exact same thing on both sides of the equal sign

Symmetric – uses the key words “if, then”If , then

Associative Property: move the parentheses around , the numbers have to stay in exactly the same order.

3x + (5x + 9) = (3x + 5x) + 97(xy) = (7x) y

Again – there is no associative property of subtraction or division

Transitive Property of Equality – if two number both equal the same thing then they equal each other (it always says if…and…then)

If , and , then

Addition Property of Equality – if you add the same thing to both sides of an equation, the two sides are still equal)

Subtraction Property of Equality – if you subtract the same thing from both sides of an equation, the two sides are still equal)

Multiplication Property of Equality – if you multiply both sides of an equation by the same thing, the two sides are still equal)

Division Property of Equality – if you divide both sides of an equation by the same thing, the two sides are still equal)

Inverse Property – any number plus its opposite = 0 ( 4 + (-4) = 0) any number times its inverse =1 (2 * ½ = 1)

Function, PatternsFunction: x does not repeat (domain is all different numbers)Graphs use vertical line test (can only cross at one point)

The coordinate plane is made up of an x-axis and an y-axis. To find a certain point on the coordinate plane, you need an x and a y coordinate that are listed together like this (x, y). The x and y axes are always labeled on the SOL.

A relation is any set of ordered pairs. The domain is the set of all x values or first coordinates and the range is the set of all y values or second coordinates. A relation is a function if all of the x-values are different. See examples below.

EX1: {(3, -4), (3, 5), (4, 6)} EX2: {(7,1), (8,1), (9,1)} EX3 EX4 not a function – the x’s repeat function

EX5: EX6:

Functions Not a Function

– Not a function, D:{3,5} The x values are repeating R: {4, 10} Not a function, the x is repeating

f(x)=abs (x+2)

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9

-8-7

-6-5-4-3-2

-1

123

456

78

9

x

y

x^2/9+y^2/25=1

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9

-8-7

-6-5-4-3-2

-1

123

456

78

9

x

y

x=3

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9

-8-7

-6-5-4-3-2

-1

123

456

78

9

x

y

function not a function – use the vertical line test not a function – use the vertical line testD = { all real numbers}R = { all real numbers greater than 0}

Slope, X, Y, Interceptsy=mx+b m=slope b=y-interceptAx + By = C standard form

Slope= m=

Horizontal line (HOY) slope = 0 equation starts with yEx. y=4

Vertical line (VUX) slope = undefined equations starts with xEx. x=4

Slope, Graph of a lineTypes of slope

Positive Negative Zero Undefined

x

y

x

y

x

y

x

y

123

456

25

8910

61218

155

4410

335

Graphing a Line

From slope intercept form (y=mx+b):

m = b= -2

From Standard Form (x and y intercepts):

1. Get y by itself (add/subtract first if x and y are on both sides)

2. Divide everything by the number in front of y

3. Identify m (slope) and b (y-intercept)

4. Start at b on your y-axis make a dot

5. Now use your slope to make at least two more dots.

6. We will go down 2 lines and over to the right 3 make another point

7. Repeat step 6 from your new point.

8. Connect your points. Place arrows at the end of your line.

x

y

x

y

x

y

1. Get x and y on the same side of the equal signs.2. To solve for x, cover up the y and rewrite the

equation. 3. Solve for x4. Your point on your graph will be at (4,0) or 4 on

the x-axis

5. To solve for y, cover up the x and rewrite the equation.

6. Solve for y7. Your point on your graph will be at (0,3) or 3 on

the y-axis

Write Linear Equations Y=mx+b slope intercept

m is your slope ( ) b is the y-intercept

Ax+By=C Standard Form

A, B, and C cannot be fractionsA cannot be negativeThe GCF between A,B, C has to be 1

From standard form to slope intercept form:

1. Move x to the other side (add/subtract)-3x -3x 2. If y has a number in front of it divide everything by that number

3. Reduce all fractions

From slope (m) and y-intercept (b):

m=2 y=mx+b 1. Start with y=mx+bb=10 y=2x+10 2. Plug in m

Plug in b

From a graph:

From slope (m) and a point:

y-intercept (b)

Rise

Run

b=2 m= =

1. Find b (where it crosses the y-axis)2. Find m (slope) (from two points start at the bottom point, count up and over to the next point. 3. Start with y=mx+b4. Plug in m5. Plug in b

x

y

x

y

8. Plot the values of x and y on your graph.

9. Connect the dots to make a line, place arrows at the end.

m =3 1. Start with y=mx+b( -4 , 2) 2. Plug in m, x, and y

3. Solve for b (add or subtract the number on the same side of b to both sides)

+12 +1214= b

4. Now use y=mx+b 5. Plug in m and b

From 2 points:

(-2, 2), (2, 10) 1. Find Slope of the two points

2. Use y=mx+b and pick ONE of your points3. Plug in m, x, and y4. Solve for b

+4 +4 6 = b

5. Use y=mx+b6. Plug in m and b

Going from Slope Intercept to Standard Form of a line:

1. Move x to the other side of the equation (add or subtract)

(x and y need to be on the same side of the equation, but they are not like terms so do NOT add/subtract them together.)

-1( ) 2. A can’t be negative, so multiply everything by -1

3( ) 3. A, B, and C can’t be fractions, if they are multiply everything

by the denominators.

To get slope from Standard Form:

Systems of Equations: 2 equations, 2 variables, 2 answers

Elimination

**now plug y=5 into either equation Solution: (-4,5)

Substitution:

Solution: ( , )

Graphing- where 2 lines intersect is the solution (x,y)

If they do not cross then they are parallel, it is no solution. If you are using substitution or elimination you will get different number on each side of the equal sign.

Ex. 4=5 (not true, therefore no solution)

If they are the same line, it is infinitely many solutions. If you are using substitution or elimination you will get the same number on each side of the equal sign.

Ex. 6=6 (true, therefore infinitely many)

On Calculator: Set each formula in standard form .MenuEQUAF1: SimultaneousF1: 2 unknowns (x and y)Type in A, B, and C for each equationF1: Solve

Monomials, Exponents, Scientific Notation

A. Multiply the coefficients first, then add exponents when multiplying polynomials

B. Multiply the coefficients by itself the number of times the exponent indicates, then Multiply Exponents when raising a polynomial to a power

C. Reduce (divide) the coefficients, then Subtract Exponents when dividing polynomials

You cannot have a negative exponent. If you get a negative exponent, move the variable and exponent to the other side and make the exponent positive

Ex: =

Scientific Notation 2.53 x = 253,000 3.06 x = 0.0000306

Polynomials Add/Subtract like terms

(The same variables with the same exact exponents)

Multiply/Distribute, FOIL (First Outer Inner Last)

Factoring Binomial/Trinomial EquationsSigns Trinomial= 2 binomials(+ +)=(+)(+) =(- +)= (-)(-) = (+ -)=(+)(-) =(- -)= (+)(-) =

Different of Squares

On calculator: Set equation equal to zero:Go to menuSelect EQUAF2: PolynomialF1: 2nd degree (highest exponent)Type in the values for A,B, and C (if there is no A, B, or C in your equation type in 0)F1: Solve

These are your roots, solutions, zeros to get the factors: place in (x___) with the opposite value.

The factors will be (x+1)(x-4)

The factors will be (4x+3)(x-2)

To turn a root into a factor:Take the opposite sign and put in (x __) after x

If you have a fraction as a root: the denominator goes in front of (_x) and the numerator takes the opposite sign and goes after x

Radicals

Simplifying RadicalsTo simplify, use n n nab a b or break out into prime factors looking for the same repeated factors (2 or 3 or 4 of a kind—depending on the index).

Ex) 2

27 3 3 3 3 3of a kind

Ex2) 3 54 Comes out of the radical

Simplifying Radicals with variables:When working with variables, divide the exponent by the index. The amount divided evenly is the exponent for the variable on the outside. The remainder, if any, is the exponent for the variable on the inside.

Ex) 2325 5 5a pair

x x x x x Ex2)

Solve Quadratic Equation Set trinomial =0 Factor Polynomial like normal Set each factor (set of parenthesis) equal to zero Solve for the variable You should have two answers!

Graphs Parabola Answers is where graph crosses the x-axis

Using the calculator: Set equation equal to zeroMenuGraphType in equation F6: DrawF5: GSOLVF1: RootUse the side arrows to go from one root to the other

FactorsRoots, Zeros, Solutions

Factors

Roots, Zeros, Solutions(Where the graph crosses the x-axis)

Using the quadratic formula: solve:

Plug A, B, C into your equation.

X=3 and x= -4

Functional Value=substitute a value in for the variable

Plug in value for x

f(x) = 3x – 5 f(x) means the same thing as y. It is just the name of the functionf(7) = 3(7) – 5 = 16 f(7) means take out x, put in 7

EX for f(x) = x2 – 1 find the range if the domain is { -2, -1, 0, 1} this means put all of those numbers in for x and get an answer for each one. Put all of the negative numbers in parentheses to square them.

Answer {-1, 0, 3} – in order from smallest to biggest. You got 0 twice but you don’t have to list it twice.

Line of Best Fit Enter the set of data into a list (or lists) on your graphing calculator. Look at the scatterplot graph, decide which model is most reasonable (linear, quadratic, cubic,

logarithmic (LN), or exponential) Calculate the appropriate regression formula using the calculator and plugging in a, b, etc. Write the equation down.

Example 1 : Given the data below, find the line of best fit. What would be the value of y when x=6? {(1, 2.1), (3, 3.1), (5, 4.0), (7, 5.2), (9, 5.9)}

Plug x into List 1 and y into List 2 Graph the scatterplot. It should represent the line .485 1.635y x

Using the equation for the line of best fit, predict the y value when x = 6: Plug 6 in for x. Simplify

.485(6) 1.635 4.545y

On calculator: MenuSTATType all values of X in list 1Type all values of Y in list 2F1: GPH1F1:GPH1F1:CALCF2: X

Replace m and b with the data given

Curve Best FitWrite Equation using

Example:The manager of a large bookstore counts the number of books that are sold each hour since the store has opened. For example, since the store opens at 8:00 A.M., she will record the number of books sold from 8:00-9:00 a.m. as the number sold in the first hour. The table below shows some of her data.

Hours Since Opening 0 4 8 12 16Number of Books 0 26 38 37 22

Write an equation for a function to model the data.

Type the Hours in L1 and Books in L2You will see the following on your calculator:

Quad Reg

a= -.4241071429b= 8.160714286c= .0285714286

=.9999396718

Plug a, b, and c back into the equation. Round to the second decimal place.

This is the equation for the curve best fit!

On calculator: MenuSTATType all values of X in list 1Type all values of Y in list 2F1: GPH1F1:GPH1F1:CALCF2: F6:draw

Statistics

Mean- averages (add up all the data and then divide by # of objects)Mode- MostMedian- middle # put in order smallest to biggestRange- subtract the lowest value from the highest valueStem-leaf, Box-whisker read directions, look at graphs

Normal Distribution:

A normal distribution shows data in a symmetrical, bell-shaped curve. Data is centered around the mean ( )The standard deviation ( ) tells how each data value in the set differs (deviates) from the mean.The variance ( ) is the squared deviation from the mean of a data set.

On the Calculator: MenuStatsType data into L1F2: CalcF1: Var

mean (μ)σx= standard deviation Label the normal curve indicating the following: mean and deviation

Example 2: The test scores on a college algebra test are as follows: 67, 69, 71, 75, 78, 78, 83, 85, 85, 85, 85, 86, 87, 89, 92, 95, 98, 98, 98, 100, 100, 100, 100, 100, 100.

A) What is the mean?B) What is the Standard deviation?C) What is the absolute mean deviation?

Z-ScoreA “z-score” represents the number of standard deviations away from the mean

A z-score with a negative value lies below the mean. A z-score of 0 lies at the mean A z-score with a positive value lies above the mean.

Z-scores are a way to compare different normal distributions,

To calculate the value of a z-score, ,xz

where is the mean and is the standard deviation. “x” is the

number you are seeking.

Example 4:The mean height of eleventh-grade boys at Franklinton High School was 69.5 inches and the standard deviation of the data was 3 inches.

The mean height of eleventh-grade girls at Franklinton High School was 64 inches and standard deviation of 2.25 inches.

How many standard deviations away from the mean is a boy who is 65 inches and a girl who is 65 inches tall?

Find the z-score of Boys: Find the z-score of Girls:65 69.5 1.5

3z

65 64 0.442.25

z

Direct Variation Equation: y = kx (k is the constant of variation); graph is a line thru the origin Solve first equation for k substitute k into another equation and solve for the unknown variable

Example 1: If y varies directly as x and y is 6 when x is 18, find y when x is 24.

Set up to solve for k: 6 18

13

y kxk

k

Then plug k into the formula and find the missing variable

1 (24)38

y kx

y

y

II Inverse Variation Example – the speed of a car and the time it takes to reach the destination

Equation: xy k or kyx

( k is the constant variation)

graph is a hyperbola in opposite quadrants (Quad I & III or Quad II and IV) To solve find k and substitute it and remaining numbers into eqn. again.

Example 2:If y varies inversely as x and y = 10 when x = 2, find y when x = 6.

If (2)(10)20

xy kk

k

Find y now

6 20206

103

xy ky

y

y

x

y

x

y