SAT Flashcards
ONE OF EVER X dollars =
1/X DOLLARS
A meteorologist estimates that on a sunny day, the air temperature decreases by about 4F for every 1,000feat of elevation gain.
On a certain day, the air temperature outside an airplane flying above Seattle is -58F∘ and the ground level temperature in Seattle is 70F . If x is the height above Seattle, in feet, at which the plane is flying, which of the following best models the situation?
70 = - 4x/1000 +58 70 = - 4x/1000 -58 70 = 4x.1000-58 70 = 4x/1000 -58
y = ax + b
b : is the initial one - the ground height is 70ft and its go up by 1000ft , while the temperature decreases 4F ( -4)
a: is the variation, slope
-58 = -4x/1000 + 70
But it does not appear like this in the answer, let’s chng the ax position
-58 +4x/1000 = -70
It is not : 70 = - 4x /1000 - 58 . Since it is temp decreases for every of ELEVATION GAIN
Oliver mows lawns in his neighborhood. He charges $10 for each regular yard he mows, and he charges an extra $5dollar sign, 5 for each large yard that he mows. In one week he mowed 6 more large yards than regular yards and made$265 If r represents the number of regular yards that Oliver mowed, which equation best models the situation? 10 ( r+6) = 265 10r + 5 ( r+6) = 265 10r + 15( r+6) = 265 10r + 15r + 6 = 265
” an extra $5dollar sign, 5 for each large yard “ = 10+5 = 15$ each large
10r + 15 ( r+ 6) = 265
Cecilia rents a moving truck that uses a gallon of gas every 10 miles for $29.95dollar sign, 29,95 per day. She incurs an additional charge of $0.89 for each mile traveled and must return the truck with the same amount of gas in the tank as when she rented it. Cecilia travels x miles and replaces the gas she used with gas that costs$2.25. She returns the moving truck on the same day, and calculates that she spent a total of $65.63Which of the following equations best models the situation?
Total = rent fee + additional mile charge + gas return ( = when she rented it)
2.25$x /10 = the gallon used
29,95$ + 0,89$.x + ( replaces the gas she used with gas that costs$2.25.) 2.25$x /10= 65.63
Camille and Hiroki have decided to start walking for exercise. Camille is going to walk 7 miles the first day and 3 miles each day after that. Hiroki is too busy to walk on the first 2 days, so he decides to walk 5 miles each day until he has walked the same number of miles as Camille. If Camille and Hiroki will have walked the same number of miles, how many days will Camille have walked?
Camile miles = Hiroki miles
Camile walked 2 days extra : 7 + 3 ( d-1)
Hiroki walked 2 days fewer: 5 (d -2)
7 + 3 (d -1) = 5 (d-2)
4 + 3d = 5d - 10
-2d = -6
d = 3 days
range :
the min should be the result of __
the max should be the result of __
On a highway, drivers are required to maintain a speed of between 55and 65miles per h.$7penalty is assessed for each 1mph a driver’s speed is outside this range. If a driver receives a 42 penalty, what are possible values for his speed?
min : substraction : 55-6 = 49
max : plus 42 = 6 = 71
42/7 = 6 mph
At the county fair, the operator of a game guesses a contestant’s weight. For each pound the operator’s guess differs from the contestant’s weight, the contestant will receive $3A contestant weighing xpounds received $15 when the operator guessed 120pounds. Which of the following equations could be used to solve for the weight of the contestant?
! If the options do not have one that has + - in the answer, it doesn’t matter if it was more weighted or less
To only know what it differs : 120 - x
When its differed , the constant earn 3$ : ( 120-x)3
At final, the constant earn a total of $15 :
( 120-x)3 = 15
A taut string of length 10 inches is plucked at the center. The vibration travels along the string at a constant rate of c inches per millisecond in both directions. If x represents the position on the string from the left-most end, so that 0≤x≤100, which of the following equations can be used to find the location x of the vibration after 0.3 milisec
1/c (x -5) =0. 3
cx - 5 = 0,3
x - 10 = 0,3c
rate x time = distance
plucked at the center = 5
c . 0,3 = ( 5 -x)
the answer does not appear like this way , so divide “c”
0,3 = (5-x)1/c
T=0.125(I−8,225)
The amount of tax, T, in dollars to be paid by an individual with an income of III between 8{,}225$ 34{,}500$ is given by the equation above. Tax is paid from the portion of income above 8,225$ and the remaining amount becomes a part of disposable income. What percentage of income above 8,225 remains after paying tax for an individual whose income falls within the specified range?
Rearrange method
m = 0,125 = 🔺y/🔺x = Taxes/Income = 12,5% of Income PAID to Taxes.
The remain is : 100% (income) - 12,5% = 87,5
C=1.25S
A factory determines that the mass of a chocolate-dipped strawberry in grams, C, is related to the mass of an undipped strawberry in grams, S as shown in the equation above. For a chocolate-dipped strawberry, what percent of its mass is chocolate?
Dipped-chocolate - Undipped = chocolate
= 1,25S - S = 0,25
Chocolate/ chocolated-dipped strawberry = 0,25/ 1,25 = 1/5 = 20%
A food truck owner has determined that her maximum revenue occurs when she sells 950sandwiches per month. For every sandwich above or below 950that she sells, her revenue decreases by $0.10Which of the following could be s, the number of sandwiches sold in a month if the owner’s revenue decreased $45 from the maximum? Round the answer to the nearest whole number.
“above or below 950 sandwiches” :
( 950-s) decreases
“VARIES , decreases by 0.10$”
0,10 ( 950-s) = 45
A stack of 20 stainless steel sheets is supposed to be 40mm thick. The allowable amount of variation in thickness (tolerance) is 0.08mm for an individual sheet. Which of the following are the smallest and largest allowable thicknesses for a stack of 20sheets?oi
smallest and the largest : (40 - x)
“is 0.08mm for an individual sheet. “: 40-x/ 20 sheets
the allowable variation is 0,08:
40-x/ 20 = 0,08
largest = 40 + x smallest = 40 - x
A company that produces thumb drives has determined that its maximum profit occurs when it sells 5000. For every 500 units above or below 5000 company sells, its profit decreases by $1500 Which of the following could be the number of units sold in a month if the company’s profit decreased $12,000 from the maximum? Round the answer to the nearest whole number.
“bove or below 5000 company sells”
( 5000 - x)
“For every 500 units…. decreases $1500”
500/1500 ( 5000-x) = 12000
From 1990 to 2010, the annual profit of a company was $350 times the numbers of years either before or after 200 less than $750. For which of the following years could the annual profit have been $575
see
In the year 2006, the average home price per square foot in a certain county was $98. For each year before or after 2006, the average price per square foot increased by approximately $3.50 In which of the following years could the average home price per square foot be $119
“each year befo or after 2006”
2006 - x
Increased by $3,5 + the already existing in 2006
(2006-x) .3,5 + 98 = 119
Uche is a cartographer. He picks a scale to fit a map of India onto a page of an atlas. The page is 12 by 12 inches, with 0.75 inch margins on all 4 sides. India measures 3,214 kilometers from north to south and 2,933 kilometers from west to east. Uche wants the longest dimension of India to fit exactly in between the margins of the page. If k is the number of kilometers per inch in Uche’s scale, which equation best models the situation?
lET'S TAKE ONLY ONE side margin 0,75 of 2 sides : 12 - ( 0,75.2) = 10,5 k = number of kilometers per inch = km/inch = 3,214 km/10,5inch
k.10,5 = 3,214
The gas mileage for Peter’s car is 21 miles per gallon when the car travels at an average speed of 50 miles per hour. The car’s gas tank has 17 gallons of gas at the beginning of a trip. If Peter’s car travels at an average speed of 50 miles per hour, which of the following functions f models the number of gallons of gas remaining in the tank t hours after the trip begins?
50miles — x
21 miles — 1gallon
y=mx + b
G= 🔺y/🔺x . t + initial
BUT, since x or t is always multipying the numerator, it should be 50t above the 21
G = 50t/21 ( miles per G) - 17
Is minus 17, cuz G is the REST of the G
Line l with equation 5x-6y= 3 is graphed in the x-y plane. Line k with equation 2x+ny=5 is perpendicular with line. What is the value of n
Perpendicular: solpe 2 is negative inverse = slope 1
Slope 1 : -A/B = -5/-6
Slope 2: 5/6 => -6/-5 = 6/5
B (n) = -A/B = 6/5 = 2/n ( multiples crossing)
6n= 10 : n = 5/3
in the x-y plane , line p has equation y=ax-4 where a is a constant. If line p intersects the x axis at x = 3,what is the value of a
WHEN LINE INTERSECTS THE X AXIS -> Y= 0
0 = a.3-4
Which of the following is an equation of the line A graphed in the xy plane that passes through the point (-1, 3.5)and is perpendicular to the line B whose equation is x + 4.5 = 0
x= 3.5
y= 4.5
y=3.5
x=-1
x = -4,5 . This means that if y=0 , the x will be always -4,5. While, if y is not 0, the y will be constant
x= -4,5 and y= 3.5
A line in the x y-plane passes through the point (4, -1)and is perpendicular to the line with equation y = x + 5y Which of the following is an equation of the line ?
x = 1.x => a=1
perpendicular => -1
y=-1x + b
(4, -1)
-1 = -1.4 + b
3 = b
y = -1x + 3
when x + 3 or y + 3 = 0 , the x or y is a ___
Which of the following is the equation of the line A graphed in the x y-plane that passes through the point (-2,2)and is parallel to the line B whose equation is y - 3 = 0
a) x + 2 = 0
b) y + 2 = 0
c) y-2 = 0
d) x-2= 0
y is constant
y = 2 => y-2=0 ( y-3=0 : y=+3)
y-2=0
What is the equation, in which its slope is 7/8 and y-intercept is 3
a) 7x+8y= 24
b) 7x - 8y = -3
c) 7x-8y= 24
d) 7x+8y = 3
e) 7x-8y= -24
slope = -A/B -> - (-7) /8
y intercept = 3 = C/B = C/8 = 24
-7X+8y = 24
HOWEVER, in the answers the “Ax” are all positive:
(-7X+8y = 24) x -1
+7x -8y = -24
If the slope is + , the line will be
m + => line rising up
x-1-y = 0 switch into the y=mx+b form
x = 1x = mx
- y ( go to the opposite) => +y = 1y
- 1 = b
\+1y= 1x - 1 0 = x-1-y
how is the graph of 4x - 10 = 0 gonna look like
there is no y , so the line will be entirely vertical line passing through the x value when f(x) = 0 :
x = +10/4 = 5/2 = 2,5
The equations x+y=3x and 5x−5y=−15 are graphed in the xy plane. Which of the following must be true of the graphs of the two equations?
a) the slope of “x+y=3” is 1 and the slope of another is -1
b) they are perpendicular
c)
a) 1/1=0 |. -A/B -1/-1=1
b
If there is no “x” or “y” in the equation question, the one that is existed will remain ___.
Which of the following is the equation of the line A graphed in the XY plane that passes through the point (-2,2) and is parallel to the line B whose equation is y - 3 = 0
Justify
a) x-2= 0
b) y+2 = 0
c) x + 2 = 0
d) y - 2 = 0
constant;
y= +3
So, there need to be :
y= 2 => 2-2= 0
It’s not x+2 = 0 -> -2+2 = 0 .
Because there will be no y.
Y exist and is constant to “2”. X is the one who does not exist
A jar of jelly bean candies weighs 8ounces. A container of caramel candies weighs 40 ounces. Lúcia buys 10pounds worth of jelly bean candy jars and caramel candy containers for her big party. Given that there are 16 ounces in a pound, which of the following equations correctly relates the number of jars of jelly bean candies, j, and the number of containers of caramel candies, c, that Lúcia bought?
8ounces —– x pounds
16 —– 1 = 8/16 ( = 40/16)
8j/16 + 40c/16 = 10pounds
A forest that is populated by crows and doves currently contains a total of 1,500 and dove nests. Crows use approximately 8 grams of twigs when building a nest, and doves use approximately 6grams of twigs when building a nest. Which of the following equations best approximates the relationship between C, the total amount, in grams, of twigs used for crow nests and D, the total amount, in grams, of twigs used for dove nests?
8g —- 1 nest
Cg —- x nest = C/8 ( = D/6)
C/8 + D/6 = 1,500 NEST
A company produces candy bags that each hold about 528cubic inches of candy. Each bag is filled with any mixture of lollipop candies and gummy bear candies. When a bag contains only lollipop candies, then it has about 361candies. When a bag contains only gummy bear candies, then it has about 697candies. Given any candy bag produced by this company, which of the following equations could relate the approximate number of lollipop candies, l, in the bag and the approximate number of gummy bear candies, g, in the bag?
A) l/361 + g/697 = 528
b) 361l + 697g= 1
c) l/361 + g/697 = 1
d) 361l + 697g= 528
c
! 528 = inches cubes = volume
the “l” and “g” and 261,697 are UNITS
1 = max capacity
l ( number existed)/ 361 ( the max lolipopo)
Jacob’s classroom contains tables and chairs, each with 4 legs. Jacob puts a pad on the end of every leg so that the tables and chairs will not make noise when they are moved around. If Jacob needs exactly 80pads for his classroom, which of the following equations correctly relates the number of chairs, c, to the number of tables, t ?
1 leg = 1 pad
4t+4c = 80pads 4c= 80-4t c = 80/4 - 4t/4 c = 20 - t
Sterling silver is an alloy of silver that is 92.5% pure silver. If x grams of sterling silver are mixed with y grams of an 88% silver alloy to produce a 91%silver alloy, which of the following equations correctly relates xand y?
A. 0.075x+0.12y = 0
B. 0.015x -0.03y = 0
C. 0.925x +0.88y = 91
D. 0.925x +0.88y = 0.91xy
X = sterling silver has 0,925g of Pure Silves Y= silver alloy has 0,88g of Pure Silver
Since X and Y is mixed together:
92,5%x + 88%y
91% is pure silver
92,5%x + 88%y = 91% (x+y)
BUT, there is not this in the alternatives -> Change to other form : = 0
92,5%x + 88%y - 91% (x+y) = 0
92,5x + 88y - 91x+91y = 0
0.015x−0.03y=0
When is f(x) mentioned to be in the equation:
Ebuka needs to take a taxi, which costs$3.50 and an additional cost of 3,5per mile. Which of the following gives the number of miles, m(d) that Ebuka can ride a taxi as a function of the amount of dollars, d that he has?
a) 3.5d + 3.5 = m(d)
b) 3.5d/3.5 = m (d)
c) 1 + d/3.5 = m (d)
d) 1 - d/3.5 = m (d)
m(d)3.5 + 3.5 = d
- 5 - d = -3.5m(d)
- 5 - d = m (d)
- 5/ 3.5 - d/3,5 = m (d)
1-d/3,5 = m (d)
When is f(x) in the equation
Surya’s fence is 340eet long. It is made from both 6‑foot fence panels and 9‑foot fence panels. Which of the following shows the number of 9‑foot fence panels, l(s), , in Surya’s fence as a function of the number of 6‑foot fence panels, s, in Surya’s fence?
6s + 9l(s) = 340
6s - 340 = 9 l (s)
6s/9 - 340/9 = l (s)
The first is not 0, and has a n < y < n2 :
Over the course of 4 years of training for the 100-meter dash, Erica’s best time at each end-of-year track meet improved linearly by 0.30 per year. Her best time at her first end-of-year track meet was 13seconds. Which of the following equations shows Erica’s best time, b after y years of training for 1
( y-1)
b = 13 - 0,3 ( y-1)
A used car dealer would like to cover their vehicles using 220 liters of paint and 400 liters of clear coat. It takes 13liters of paint and 9 liters of clear coat to cover one car. It takes 1.5 times those amounts to cover one truck. Which of the following shows f(c) the maximum number of trucks the dealer can cover if the dealer covered c cars?
a) f(c)= 220 - 13(1,5)c / 13
b) f (c) =220-13c/13.1,5
c) f(c) = 220-9c/13
f(c) = number of trucks
13c + 13.1,5.f(c) = 220
13.1,5 f (c) = 220 - 13c
f(c) = 220 -13 c/ 13.1,5
A construction company is going to build two styles of houses in a particular area. The company will build x houses that each require 4,000 bricks and y houses that each require 8,000 bricks. For the construction project, the company ordered 350,00, so that as many as 4%, percent can break and there will still be enough for the houses. Which of the following equations expresses y in terms of x?
y= 2x + 350,00.0,96/8000 y = 1x/2 - 350,00.0,96/8000 y= 2x - 350,00.0,96/8000
4000x + 8000y = 350,00 . ( 100% - 4%)
350,00. 0,96 - 4000x = 8000y
350,00. 0,96 - 4000x/8000 = y
350,00.0,96/8000 - 1x/2 = y
In tennis, a player must win at least 6 games to win a set. If g is the number of games the player won and s is the number of sets the player won, which of the following inequalities must be true?
6g < = s
6s < = g
6s > = g
6g > = s
s < = 6g
s > = 6g
6s < = g
” AT LEATS X TO ( or for) y “ :
g/ s = 6/1
1g = 6s
6s < = g
( Inequality) “At most” = ___
A sundae requires 3] ice-cream scoops and 4strawberries, and a milkshake requires 2 ice-cream scoops and 6 strawberries. Ramses wants to make sundaes and milkshakes with at most 25 ice-cream scoops and 37 strawberries. Let’s form a system of inequalities to represent Ramses’s conditions. Let S denote the number of sundaes he makes and M the number of milkshakes he makes. Which system of inequalities best describes this situation
at most = at least
3S + 2M < = 25
4S + 6M < =
Elena is designing a paint can with thickness t millimeters and height h centimeters. She calculates that the thickness of the can in millimeters must be at least 0.10 times the height of the can in centimeters in order to withstand pressure. Due to cost constraints, the cost of material used, (0.2 + t + 0.5h) cents, must be at most 12.2. Which of the following systems of inequalities best models the relationship
between height and thickness described above?
t < = 12 - 0,5h
10t > = h
t > = 12 - 0,5h
10t < = h
t > = 12 - 0,5h
t < = 10 h
“thickness at least 0.10 x h”
(mm) t > = 0.10h ( cm).10
Since we are using “h” as x, we should :
10t = h
(0.2 + t + 0.5h) < = 12,2 ( at most = at least)
t < = 12,2 - 0,5h - 0,2
t < = 12 - 0,5h
At least x for each y =
A barge must carry steel pieces for a construction project such that the total weight of the steel pieces is under 1,500kg. Each steel piece is either a beam, which weighs 363kg. There must be at least 2 connector plates for each beam. If b is the number of beams and c is the number of connector plates, which of the following systems of inequalities must be true?
a) 363b + 6c < = 1500
2c < = b
b) 363b + 6c < 1500
2c > = b
c) 363b + 6c < 1500
2b < = c
d) 363b + 6c < 1500
2b > = c
“AT LEAST 2c FOR EACH b’ =
c/b > = 2/1
1c > = 2b
d
A taxicab company charges $2.05 for the first mile, $1.56for every successive mile after that. Which of the following functions best represents the amount charged, in dollars, by the taxicab company after m miles (m > 4)
a) f(m)=2.05+1.56(m−4)
b) f(m) = 3,61 + 1;22 (m-4)
c) f (m) = 6.73 + 1.22 (m-4)
d) f(m) = 7.21 + 1.22 (m-4)
c) 2.05 + 1.56x3 miles = 6.73 + 1.22 (m-4)
At t it has x , then, at t2 it has x2 :
Snow from the most recent storm fell at a constant rate for the 10 hours the storm lasted. Three hours into the storm, there were 8.48 inches of snow on the ground. Six hours into the storm, there were 10.5 of snow on the ground. If it has been thours since the start of the storm, which of the following best approximates the amount of snow, S in inches, that is on the ground?
S(t)=2.8t
S(t)=3.5t
S(t)=0.7t+6.3
(3, 8.48) and ( 6, 10.5)
m = 10.5 - 8.48 / 6-3 = 2.1/3 = 0.7
S(t) = 0.7t + b 8.48 = 0.7x3 + b = 6.3
S(t)=0.7t+6.3
A physicist measures the energies of atoms A and B to be the positive values a and b respectively. He determines that the energy of atom B is between one percent more than and one percent less than twice the energy of atom A. Which of the following systems of inequalities best models the possible range of values for the energies of atoms A and B?
a<2.02b
a>1.98b
a<2b+0.02
a>2b−0.02
B > 2A x 1,01
B < 2A x 0,99
B> 2,02a
B < 0,98a
The supply and demand curves for a product are given by functions Sand D, respectively. For a given number of units, Q, S is the minimum price, in dollars, that the supplier should accept. D is the maximum price, in dollars, where consumers will purchase that number of units.
S(Q) = 45 + 0.04Q
D(Q) = 110 - 0.003Q
For the above equations, which of the following combinations of price and quantity fall between the maximum consumer price and minimum supplier price?
500 units at $105
1000 units at , 80
1500 units at $ 110
2450units at 110
Replace each answer to the equations. The value should be more than S and less than D
A business analyst is deciding the amount of time allotted to each employee for meetings and training. He wants the sum of meeting and training time to be no more than 16 hours per month. Also, there should be at least one training hour for every two meeting hours. Finally, there should be at least 2 meeting hours per month to discuss short-term goals. What is the difference between the maximum and minimum number of monthly training hours that could be allotted to an employee? A)10 B)13 C)14 D)1
“t least one training hour for every two meeting hours. “ :
1training < = 2meeting
t < = 1/2
m + 1m /2 < = 16hour
the max = 16 hour
the min is 2 meeting hour + 2.1/2 = 3
A study by a group of cardiologists determined that maximal heart rate in humans, a parameter used as a basis for prescribing exercise programs, depends on age. Maximal heart rate is around 201 beats per minute for a 10 years old, and decreases at a constant rate of 7bpm every 10 years. Which of the following inequalities best describes the ages, a, for which maximal heart rate is less than 180bpm for a≥10
208−0.7a<180
208−0.7a≥180
201-0.7a<180
201−0.7a≥180
- 7bpm/ 10 years = variation = m
initial stage = 201 ( 10years old) = b
201-0.7a<180
Joanne and Richard volunteer at a hospital. Joanne volunteers 4 hours more per week than Richard does. In a given week, they do not volunteer for more than a combined total of 16 hours. If x is the number of hours that Richard volunteers, which inequality best models this situation?
x+4≤16
2x+4≤16
2x+8≤16
2x−4≤16
Richard(x) = J - 4
J + x = 16
x+4 + x <16
2x+4≤16
From 2005 to 2011, the world’s forests were decreasing in area by about 55.776
per year. In 2008
240,351,492 km squared of forest land. If x represents years since 2005, which of the following best predicts the area of the world’s forests, a, for 0< a < 60
” x years since 2005” ( x - 2005)
If 2008 was 240.351, so 2005 was :
240.351 = ( 2008 - 2005). 55.776 - initial value
initial value = 73023
y = mx+ b y = 55.776 ( x-2005) - 73023
a is constant and there are infinitive solution :
What is the value of a
a( y-1/3) + x/2 =0
3y - x -1 = 0
1) Convert to standard form :
y = mx+b
2) Find the slope
3) If “a” is a constant, and INFINITIVE solution, then both slopes are equal
1) ay - a/3 = -x/2
y = (-x/2 + a/3 )/ a
y = (-x/2a + 1/3 )
y = +1/3 + x/3
2) slope = xm
- 1/2a .x and 1/3 . x
3) - 1/2a = 1/3
- 3 = 2a
- 1,5 = a
**a is a constant and there are NO SOLUTIONS “
ay = 2x + 1 y = 2x + 2
1) Find the slope
2) Parallel = no solutions = same slope
y = 2x / a + 1/a 2x = 2x/a ( false - so it's no solution)
a = 1
0.5(8w+2v) = 3
8w =2−v+4w
Which of the following accurately describes all solutions to the system of equations shown above?
a) v=1 , w = 1/4
b) v=2 , w = -1/4
c) infinitive solution
d) no solution
4w + 1v = 3
4w + v = 2
FALSE ! => 0 SOLUTION
“How many solutions” :
1.70p−0.34q = 0
0.17(q+1)−0.85(p−1) = 0
Consider the system of equations above. How many (p, q)( solutions does this system have?
1) Convert into standard form
2) find the slope
3) compare two slope
! (p,q) = (x,y)
q = -1.70p / 0.34
q = -1.70p / 0.34 - 1.02/ 0.17
2) -1.70p / 0.34 and -1.70p / 0.34
3) We have SAME SLOPES but y-intercept DIFFERENT => PARALLEL = 0 SOLUTIONS OR IFINITIVE
4) Replace the m and compare 2 equations
5p - 0.12 = q
5p = q
INCOHERENT - 0 SOLUTIONS
44(j+2k) = 12
22k =−11j+16
** How many solutions:*
Consider the system of equations above. How many solutions (j,k)does this system have?
How many solutions : convert to y=mx + b and FIND THE SLOPE
44(j+2k) = 12 88K = 12 - 44j k = 12/88 - 44j /88
22k =−11j+16
k = -11j/22+ 16 /22
- 44j /88 (½) = -11j/22
Ifinitive solutions
19−38y = 76 x
24x =−6(2y−1)
**How many solutions ** :
Consider the system of equations above. How many solutions (x,ydoes this system have?
How many solutions : convert to y=mx + b and find the SLOPE
19−38y = 76 x
24x =−6(2y−1)
y = 76x/-38 - 19/-38
+12y = -24x/12 + 6/12
- 24x/12 = 76x/-38
- 2x = -2x
There are IFINITIVE solutions
3x+2y =4(x−y−6)
6(y+x) =7x−24
Which of the following accurately describes all solutions to the system of equations shown above?
Since in the answers there are “ifnitive solutions “ “No solutions”, let’s identify the SLOPE first
1) Isolate y :
3x+2y =4(x−y−6)
6y = x - 24
y = x/6 - 4
6(y+x) =7x−24
6y = 6x - 24
y = x/6 - 4
2) Same slope : x/6 = x/6
IFINITIVE SOLUTIONS
12t =4v−3
−6t=4v+6
If (t, v) is the solution to the system of equations above, what is the value of t - v?
12t - 4v = 3 (.+1)
−6t - 4v =+6
6t = 9
t= 3/2
-6 ( 3/2) - 4v = 6
-4v = 6 + 9
v = 15/4
3/2 - 15/4
6/4 - 15/4 = -9/4 ❌
6/4 > - 15/4 => +-9/4
3−m=2(ℓ−4)
m=ℓ−4
Consider the system of equations above. If (ℓ,m) is the solution to the system, then what is the value of l ?
3−m=2 . m => m=ℓ−4
3 = 2m + m
3 =3m
1= m
1 = l - 4 l = 5
When the 2 equations are equivalent:
−20x+12y = 24
−5x+3y = 6
Consider the system of equations above. How many (x,y)solutions does this system have?
2 equations ‘re equivalent => there are IFINITIVE solutions
1) Isolate f(x) ,y
2) Do the system of addition
24 + 20x = 12y
6 + 5x = 3y (x4) => 24 + 20x = 12y
x² + 13x/2 + 15/2 = 0
If x = a and x = b are the solutions to the equation above, hat is the value of ab
! ab is LITERALLY the product of ab ( the roots )
DON’T NEED TO FIND ROOTS
ab = c/ a
15/2 /x
15/2x
( x + 3)² - 4 = 0
Find the roots
(x + 3 ) ( x+3) ❌
3.3 = 9 - 4 = 5
3+3 = 6
Sum 6 = 1+5
Product: 5 = 1.5
the roots are : INVERT THE SIGNALS
-1 and -5
the one with “x” is the __ of the roots.
The one with no “x” is the __ of the roots
Describe each formulas
with “x”: Sum = -b/a
no “x” : Prod = c/a
Sara has whole milk, which contains 3.25% butterfat by volume , and one type of low - fat milk, which contains 1% butterfat by volume. Which of the following systems could be used to determine the amount of whole ,ilk,w, in ounces, and the amount of low-fat milk, l, in ounces, that Sarah should mix to obtain 32 ounces of a low-fat milk with 2% butterfat by volume
a) l + w =
__ l + ___ w =
l ounces + w ounces = 32 ounces
1% = 0.01
0.01l + 0.325w = total of % of the final volume
0;01l + 0.325w = 32.0.02
In 1960. the population of Colombia was about 16 million and the population of Argentina was about 20.6 million. Between 1960 and 2010the population of Colombia increased by about 530,000 people per year, and the population of Argentina increased by about 340,000 people per year. Let P represent the population, in millions, and t represent the number of years since 1960. Which of the following system of equations can be used to find the year when both countries had approximately the same population?
P =
P =
Unit = number million 530,000 = 0.53 million 340,000 = 0.43 million
Poplutation the same :
P = 16 + 0.53.t
P = 20.6 + 0.43 .t
