ALGEBRA 1 MID YEAR STUDY GUIDE
4 STEPS OF PROBLEM SOLVING
1. EXPLORE-GENERAL UNDERSTANDING OF THE PROBLEM AS TO WHAT IS GIVEN AND NEEDED
2. PLAN- STEPS NEEDED AND OPERATIONS NEEDED TO SOLVE
3. SOLVE- CARRY OUT THE METHOD
4. EXAMINE – DOES THE ANSWER MAKE SENSE COMPARE TO AN ESTIMATE
Order of Operations:
1. First do all operations that lie inside parentheses.
2. Next, do any work with exponents or radicals.
3. Working from left to right, do all multiplication and division.
4. Finally, working from left to right, do all addition and subtraction.
Property Laws
#1. Commutative properties
the commutative property of addition states that numbers can be added in any order and it will not change the sum. The commutative property of multiplication states factors can be multiplied in any order without changing the result.
Addition
5a + 4 = 4 + 5a
Multiplication
3 x 8 x 5b = 5b x 3 x 8
#2. Associative properties
The associative property of addition numbers can be grouped to form a sum in any and still get the same answer. The associative property of multiplication numbers can be grouped in any way to form a product and still get the same answer.
Addition
(4x + 2x) + 7x = 4x + (2x + 7x) Multiplication 2x2(3y) = 3y(2x2)
Additive Inverse – if add a positive and negative of the same number will cancel to zero
7 + (- 7) = 0
Multiplicative Inverse- if multiply a number by its flip will equal a positive one
1/4 x 4/1 = 4/4 =1
#3. Distributive property
the distributive property involves both addition and multiplication. A longer name for it is, "the distributive property of multiplication over addition." The term is multiplied by terms in parenthesis, we need to "distribute" the multiplication over all the terms inside. 2x(5 + y) = 10x + 2xy
#5. Identity property
the identity property zero added to any number is the number itself. Zero is called the "additive identity." The identity property for multiplication, the number 1 multiplied times any number gives the number itself. The number 1 is called the "multiplicative identity."
Addition 5y + 0 = 5y
Multiplication 2c × 1 = 2c
Substitution- a math sentence in which the variable is replaced with a number that results in the a true sentence or in solving an equation in which an operation is replaced by a number when the operation can not be explained by an existing property law. ( 12- 3 ) + 20/5 = 9 +4 + 13
Reflexive – when the left side = the right side of an equation x + 2 = x + 2
Symmetric – when the left side = the right by operations and uses the words if and then
If 6 + 7 = 13 then 13 = 6 + 7
Transitive- when 2 equation operations are equal and uses the if and then
If 6 + 7 = 13 and 4 + 9 = 13 Then 6 + 7 = 4 + 9
THE FACTS ABOUT INTEGERS
1. ABSOLUTE VALUE
The absolute value of x, denoted "| x |" (and which is read as "the absolute value of x"), is the distance of x from zero. This is why absolute value is never negative; absolute value only asks "how far?” not "in which direction?". This means not only that | 3 | = 3, because 3 is three units to the right of zero, but also that | –3 | = 3, because –3 is three units to the left of zero.
2. If the absolute value has a negative sign in front of the absolute value bars it means the absolute value is still positive but was multiplied by a negative one so the solution is a negative -| 3 | = -3
3. Adding Integers
If both numbers are the same sign add the total and keep the sign the same
7 + 7 = 14 (-2) + (-3) = -5
If one number is positive and the other is negative positives and negatives will cancel each other out and the sign is the number with the largest absolute value
7 + (-4) = + + + + + + + plus - - - - If the positive and negatives cancel 3 positives are left and 7 has a larger absolute value than -4
(+ + + +) + + + plus (- - - -) in () means canceled and leaves 3 positives
4. Multiplying and Dividing Integers
2 negatives equals a positive even signs = positive uneven sign = negative the absolute value of the number does not have any meaning
(-5) (-2) =10 (5) (-2) = -10 (-2) (5) = -10 (5) (2) =10
12/4 = 3 -12/4 = -3 12/-4 = -3 -12/-4 = 3
5. Subtracting Integers
If you have a positive and subtract a positive same as basic math operations
7 – 2 = 5
If you have a negative minus a negative 2 negatives make a positive changes the operation to addition
7 – (-2) = 7 + 2 = 9
If you have a -7 -3 = -10 had 7 negatives and took 3 more away = -10
EQUATION IS A MATHEMATICAL SENTENCE IN WHICH A VARIABLE IS A PART OF THE SENTENCE THAT CAN BE REPLACED WITH ONLY ONE NUMBER THAT WILL MAKE THE SENTENCE TRUE, TO FIND THIS NUMBER USE INVERSE OPERATIONS
EX. X + 5 =12 TO SOLVE SUBTRACT 5 FROM BOTH SIDES WHICH WILL RESULT IN X = 7
X – 10 = 8 TO SOLVE ADD 10 TO BOTH SIDES WHICH WILL RESULT IN X = 18
2X = 12 DIVIDE BOTH SIDES BY 2 X= 6
INEQUALITY – A MATHEMATICAL SENTENCE THAT USES AN ORDER SIGN OF > < OR > < SOLVED THE SAME AS EQUATIONS BUT WILL RESULT IN MORE SOLUTIONS, THE SOLUTION MAKE NOT BE USED FOR < > ONLY BUT MAY BE USED WITH > < TO GRPAH THE ANSWER ON A NUMBER LINE FOR < > USE AN OPEN CIRCLE FOR > < USE A CLSED CIRCLE TO SHOW CAN USE THAT NUMBER
IN SOLVING INEQUALTIES IF THE VARIABLE HAS TO BE MULTIPLIED OR DIVIDED BY A NEGATIVE MUST FLIP THE ORDER SIGN
Solve the inequality and graph the solution set.
Graph:
/ *Inv. of mult. by -3 is div. by -3,
so reverse inequality sign
*Visual showing all numbers greater than or = -3 on the number line
RATIO- A COMPAIRSON OF 2 NUMBERS BY DIVISION
PROPORTION- 2 RATIOS THAT HAVE EQUAL CROSS PRODUCTS, ARE A MULTIPLE OF EACH OTHER OR WILL REDUCE TO EQUAL FRACTIONS
MEANS-EXTREMES PROPERTY OF PROPORTIONS- PRODUCT OF EXTREMES WILL EQUAL PRODUCT OF MEAN SO A/B = C/D THEN AD WILL = BC : 2/3= 6/9 ( 2 x 9 ) =18 & ( 3 x 6 ) =18
PERCENT OF CHANGE
THE DIFFERENCE BETWEEN OLD AND NEW DIVIDED BY THE OLD TIMES 100
IF AN ITEM SOLD FOR $50.00 AND NOW SELLS FOR $80 THE DIFFERENCE BETWEEN 80 AND 50 IS 30 SO THEN DIVIDE THE 30 BY THE ORIGINAL SALE PRICE OF $50.00 WHICH RESULTS IN A DECIMAL OF .60 THAT TIMES 100 GIVES A 60% INCREASE
SOLVING EQUATIONS AND FORMULAS
USE THE LAWS OF ALGEBRA AND ARRANGE AN EQUATION TO SOLVE FOR A GIVEN VARIABLE
EX. AX - B = C SOLVE FOR X ADD B TO BOTH SIDES WHICH GIVES
AX = C + B THEN DIVIDE BOTH SIDES BY A WHICH RESULTS IN
X = (C + B) / A
Weighted Averages- word problems that focus on 2 things combined to create a third
MIXTURE PROBLEMS- WHEN 2 OR MORE PARTS ARE COMBINED TO MAKE SOMETHING NEW
AN EQUATION WILL FORM IN THE FORM OF A + B = C THE COMBINATIONS ARE USUALLY BY PERCENTAGE WEIGHTS OR BY A PRICE
THE INFORMATION IS ORGANIZED IN A TABLE WHICH LEADS TO THE INPUT OF AN EQUATION
WEIGHTTED AVERAGE – A SET OF DATA IN WHICH THE SUM OF THE DATA IS DIVIDED BY THE TOTAL OF ALL THE SUM OF THE WEIGHTS OF THE DATA
UNIFORM MOTION – ONE OBJECT MOVES AT A CERTAIN SPEED OR RATE AND USES THE FORMULA D = R x T D = DISTANCE R = RATE T = TIME
CASE 1
IF MIXED FRUIT SELLS FOR $5.50 PER POUND, HOW MANY POUNDS OF MIXED NUTS THAT SELLS FOR $4.75 SHOULD BE MIXED WITH 10 POUNDS OF DRIED FRUIT TO MAKE A MIXTURE THAT SELLS FOR $4.95
WHAT HAVE / AMOUNT (POUNDS) / PRICE(POUNDS) / TOTAL PRICEDRIED FRUIT / 10 / $5.50 / 5.5 (10)
MIXED FRUIT / X / $4.75 / 4.75 ( X )
MIXTURE / X + 10 / $4.95 / 4.95 (X + 10)
EQUATION
5.5 (10 ) + 4.75 (X) = 4.95 (X + 10 )
A + B + C
5.5 (10 ) + 4.75 (X) = 4.95 (X + 10 )
55 + 4.75 X = 4.95 X + 49.5
4.75 X – 4.95 X = 49.5 – 55
- .20 X = 5.50
DIVIDE BOTH SIDES BY - .20 X = 27.5
CASE 2 MIXTURE PROBLEMS WITH PERCENTS- IN ORDER TO SOLVE CHANGE THE PERCENT BACK TO THE DECIMAL FORM IN ORDER TO DETERMINE THE PART NEEDED IN THE MIXTURE
IF A 30 % SOLUTION OF COPPER SULFATE IS NEEDED AND YOU HAVE 40 MILLILITERS OF A 25% SOLUTION, HOW MANY MILLILITERS OF A 60% SOLUTION SHOLUD BE ADDED TO OBTAIN A 30% SOULTION?
WHAT HAVE / AMOUNT IN MILLILITERS / TOTAL AMOUNT25% SOLUTION / 40 MILLILITERS / .25 (40 )
60% SOLUTION / X / .60 ( X)
30 % SOLUTION / X + 40 / .30 (X + 40 )
.25 (40) + .60 ( X) = .30 ( X + 40)
10 + .60 X = .30 X + 12
.60 X - .30 X = 12 – 10
.30 X = 2 DIVIDE BOTH SIDES BY .30 X = 6.67 MILLILITERS
Uniform Motion- trains cars etc. if go opposite directions subtract the same divide and the distance formula is applied d = rate x time
UNIFORM MOTION
SUE DRIVES TO HER AUNT’S HOUSE AT 45 MPH RETURN TRIP IS 2 HOURS WHAT WAS HER AVERAGE ROUND TRIP SPEED?
45(1) + (22.5 ) ( 2) = 90/3 = 30 MPH
IF A CAR AND AN AMBULANCE ARE HEADING TOWARD EACH OTHER , IF THE CAR SPEED IS 30 MPH OR 44 FEET PER SECOND AND THE AMBULANCE SPEED IS 50 MPH OR 74 FEET PER SECOND, IF THE VECHILES ARE 1000 FEET APART HOW MANY SECONDS BEFORE THE CAR DRIVER WILL HEAR THE SIREN? ( THE SIREN CAN BE HEARD 440 FEET AWAY)
AMBULANCE ------CAR
1000 – 440 = 560 FEET
WHAT YOU KNOW / RATE / TIME / D = R x TCAR / 44 / T / D = 44 x T
AMBULANCE / 74 / T / D = 74 x T
44 T + 74 T = 560
118 T = 560 DIVIDE BOTH SIDES BY 118 T = 4.75 SECONDS
GRAPHING LINEAR EQUATIONS
LINEAR EQUATION- ANY EQUATION IN WHICH THE X & Y VALUES OF A PAIR WHEN REPLACED IN THE EQUATION BOTH SIDES ARE EQUAL AND EVERY PAIR THAT ALSO MATCHES THE RULE WHEN CONNECTED ON A GRAPH CREATES A STRAIGHT LINE
The solutions to the equations are called the coordinates the relation x = domain & y = range
The ordered pairs can be showed in a table list or a map, the advantage of a map is a repeated x or y is only stated once but with multiple connecting lines.
SOLVING 2 STEP EQUATIONS GOAL MOVE THE VARIABLES TO ONE SIDE AND NUMBERS TO THE OTHER TO SOLVE FOR X WITH INVERSE OPERATIONS
Function- every x can only produce only one y in that equation so x can not be repeated on a graph with y values and be a function of one equation the pairs must pass the vertical line test
Slope- the change in y / change in x rise /run on a graph of a linear equation in order to move from one pair to the next the y must go up or down and the x over in a poitive direction or backwards in a negative direction
4 types of slope
Positive – the line moves from quadrant 3 to one and the x and the y are both positive or negative
Negative – the line runs from quadrant 4 to two either the x or the y is negative
Zero Slope – these are the y = 3 y = -2 the y never changes the horizontal line of the y value
Undefined slope – these are the vertical lines of the x value the x =3 x=-4
Slope and Direct Variation reflect the changes to x that create y and explain the pattern of a graph
Direct Variation
In math when x & y are proportional in such a way that one of them is a constant multiple of the other.
y is directly proportional to x. Y = KX always passes the point of origin
Direct Variation functions the same as slope in an equation when graphing the solutions of an equation M & K are both constants- constants a number that does not change
If y = kx if y = 4 when x = 2 then if x = 16 what is y
Y = 2x so y = (2)(16) y = 32
This can also be calculated using ratio proportions of 4/2= y/16 so (4)(16) = 2y divide both sides by 2 y = 32
Direct Variation Equations
2pr or pd
A= pr2
D= rt
SLOPE- THE CHANGE IN Y OVER THE CHANGE IN X THE PATTERN OF X & Y IN AN ORDERED PAIR TABLE
The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two points on the line. Given two points (x1,y1) and (x2,y2) on a line, the slope m of the line is