Suppose that you have at your disposal a software package that solves a square nonsingular system of equations Cx=d. Assume that the package either finds the unique solution of Cx=d or terminates with an error message if C is singular. Given an m×n matrix A of rank n and an m -vector b, briefly describe in words how you would use the package to find whether or not the equations Ax=b are compatible, and if they are, compute a solution x. Comment briefly on the uniqueness of a solution if one exists.
Answers
The equations Ax = b are not compatible.
What do you mean by equation?It is represented by an equal sign (=) and is used to solve for unknown values or to describe relationships between variables. Equations can be simple or complex, involving basic arithmetic operations or advanced mathematical concepts like trigonometry and calculus.
An equation can be used to solve for a single unknown value, for example, the equation 2x + 3 = 7 can be solved for x by subtracting 3 from both sides, giving us 2x = 4 and then dividing both sides by 2, giving us x = 2. In this example, x = 2 is the solution to the equation.
To determine whether the equations Ax = b are compatible and to compute a solution x, you would first form the augmented matrix [A|b]. Then, you would perform row operations on [A|b] to reduce it to an upper triangular matrix [C|d]. Finally, you would use the software package to solve the square system of equations Cx = d.
If the software terminates with an error message, it means that C is singular, and the equations Ax = b are not compatible. If the software finds the unique solution of Cx = d, it means that the equations Ax = b are compatible and that the solution x is unique. However, it is important to note that the uniqueness of the solution depends on the rank of A being equal to n. If the rank of A is less than n, the equations are not compatible, and if the rank of A is greater than n, the equations have infinitely many solutions.
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Related Questions
The area of a rectangular room is 750 square feet. The width of the room is 5 feet less than the length of the room. Which equations can be used to solve for y, the length of the room? Select three options.
NEED ASAP
Answer choices:
y(y + 5) = 750
y2 – 5y = 750
750 – y(y – 5) = 0
y(y – 5) + 750 = 0
(y + 25)(y – 30) = 0
Answers
The equations that can be used to solve for y in the given situation are:
(A) y(y + 5) = 750
(B) y² – 5y = 750
(D) y(y – 5) + 750 = 0
What is a rectangle?An enclosed 2-D shape called a rectangle has four sides, four corners, and four right angles (90°).
A rectangle has equal and parallel opposite sides.
A rectangle has two dimensions—length and width—because it is a two-dimensional form.
The rectangle's longer side is its length, while its shorter side is its breadth.
So, the area formula of the rectangle:
750 = l * w
750 = x * (x-5)
750 = x² - 5x
x² - 5x - 750 = 0
Equations in the options that are similar:
(A) y(y + 5) = 750
(B) y² – 5y = 750
(D) y(y – 5) + 750 = 0
Therefore, the equations that can be used to solve for y in the given situation are:
(A) y(y + 5) = 750
(B) y² – 5y = 750
(D) y(y – 5) + 750 = 0
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Correct question:
The area of a rectangular room is 750 square feet. The width of the room is 5 feet less than the length of the room. Which equations can be used to solve for y, the length of the room? Select three options.
NEED ASAP
Answer choices:
a. y(y + 5) = 750
b. y2 – 5y = 750
c. 750 – y(y – 5) = 0
d. y(y – 5) + 750 = 0
e. (y + 25)(y – 30) = 0
2 FOR 2.20 UNIT RATE
Answers
Answer: (if your asking for the unit rate of that)
the unit rate is 1.10
Answer:
1.10
Step-by-step explanation:
2 /2 = 1
20/ 2 = 10
1.10 is the unit rate
In developing a new gasoline additive, researchers randomly select 10 cars and drive them both with and without the additive. The sample mean difference in gas mileage (mpg with additive - mpg without additive) is 0.41 mpg with a sample variance of 0.16. Assume the differences are from an approximately normal distribution. We want to test the hypothesis that the fuel additive has mean mpg less than the mean mpg without the additive. Calculate the test statistic.
Answers
Answer:
The test statistic is t = 3.24.
Step-by-step explanation:
We want to test the hypothesis that the fuel additive has mean mpg less than the mean mpg without the additive.
This means that the null hypothesis is that the difference is less than 0, while the alternate hypothesis is that the difference is 0 or more.
The test statistic is:
\(t = \frac{X - \mu}{\frac{s}{\sqrt{n}}}\)
In which X is the sample mean, \(\mu\) is the value tested at the null hypothesis, \(s\) is the standard deviation of the sample and n is the size of the sample.
0 is tested at the null hypothesis:
This means that \(\mu = 0\)
Randomly select 10 cars
This means that \(n = 10\)
The sample mean difference in gas mileage (mpg with additive - mpg without additive) is 0.41 mpg with a sample variance of 0.16.
This means that \(X = 0.41, s = \sqrt{0.16} = 0.4\)
Calculate the test statistic.
\(t = \frac{X - \mu}{\frac{s}{\sqrt{n}}}\)
\(t = \frac{0.41 - 0}{\frac{0.4}{\sqrt{10}}}\)
\(t = 3.24\)
The test statistic is t = 3.24.
Given that ∅ = 12.2° , calculate the area of the triange below
give your answer to 2 d.p.
Answers
Answer:
A = (1/4)√(4 + 11 + 14)√(-4 + 11 + 14)√(4 - 11 + 14)√(4 + 11 - 14)
A = (1/4)√29√21√7
= about 16.32 mm²
Answer:
16.27 mm² (see comment)
Step-by-step explanation:
You want the area of a triangle with side lengths 11 mm and 14 mm, and the angle between them 12.2°.
AreaThe area is given by the formula ...
A = 1/2ab·sin(C)
A = 1/2(11 mm)(14 mm)·sin(12.2°) ≈ 16.27 mm²
The area of the triangle is about 16.27 square millimeters.
__
Additional comment
If you use Heron's formula for the area from the three side lengths, you find it is about 16.32 mm². That's the trouble with over-specified geometrical figures. The result you get depends on which of the given values you use. (To get the area accurate to 4 sf, the angle needs to be specified to 4 sf: 12.24°.)
s = (4 +11 +14)/2 = 14.5
A = √(s(s -a)(s -b)(s -c))
A = √(14.5(14.5 -4)(14.5 -11)(14.5 -14)) = √(14.5·10.5·3.5·0.5) = √266.4375
A ≈ 16.32
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Let S be the universal set, where:
S= {1, 2, 3,..., 18, 19, 20}
Let sets A and B be subsets of S, where:
Answers
Answer:
Step-by-step explanation:
Therefore, the height of the tower is approximately 121.4 meters.
Select all lengths that are equal to 3 yards 16 inches.
Answers
The lengths equal to 3 yards 16 inches are 3 yards, 108 inches, 3.44 yards (approximately), and 108.44 inches (approximately).
To determine the lengths that are equal to 3 yards 16 inches, we need to convert the measurements into a consistent unit. Since both yards and inches are units of length, we can convert the inches into yards or the yards into inches to find the equivalent lengths.
1 yard is equal to 36 inches (since 1 yard = 3 feet and 1 foot = 12 inches).
Therefore, 3 yards is equal to 3 * 36 = 108 inches.
Now, we can compare 108 inches to 3 yards 16 inches.
108 inches is equal to 3 yards, so it matches the given length.
To convert 16 inches into yards, we divide it by 36 since 1 yard = 36 inches. 16 inches / 36 = 0.44 yards.
Therefore, 3 yards 16 inches is equivalent to:
3 yards
108 inches
3 yards 0.44 yards (or approximately 3.44 yards)
108 inches 0.44 yards (or approximately 108.44 inches)
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Find the coordinates of the midpoint M and the distance between (-5, 6) and (3, 2).Round to the nearest tenth if necessary.The coordinates of the midpoint M are____The distance between the two points is about ____See image for more clear
Answers
Explanation
We are given the following points:
\(\begin{gathered} (-5,6) \\ (3,2) \end{gathered}\)We are required to determine the midpoint M and the distance between the given points.
We know that the midpoint of two points is given as:
\(Midpoint=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})\)\(\begin{gathered} (-5,6)\to(x_1,y_1) \\ (3,2)\to(x_2,y_2) \\ M=(\frac{-5+3}{2},\frac{6+2}{2})=(\frac{-2}{2},\frac{8}{2}) \\ M=(-1,4) \end{gathered}\)Hence, the midpoint is:
\(M=(-1,4)\)Also, the distance between two points is given as:
\(Distance=\sqrt{(y_2-y_1)^2+(x_2-x_1)^2}\)\(\begin{gathered} (-5,6)\to(x_1,y_1) \\ (3,2)\to(x_2,y_2) \\ Distance=\sqrt{(2-6)^2+(3-(-5))^2} \\ Distance=\sqrt{(-4)^2+(3+5)^2}=\sqrt{16+64} \\ Distance=\sqrt{80}=4\sqrt{5}\text{ }units \end{gathered}\)Hence, the distance is:
\(D=4\sqrt{5}\text{ }units\)How do I uses these
Answers
Answer:
To understand a graph you need to interpret a graph or a chart, read the title, look at the key, read the numbers and lables. then study the graph to understand what it shows. read the title graph or chart. the title tells what information is being displayed.
Step-by-step explanation:
a pharmaceutical salesperson receives a monthly salary of $5000 plus a commission of 7% of sales. write a linear equation for the salesperson’s monthly wage W in terms of monthly sales S. NEED TO TURN IT IN TODAY
Answers
Equation for the monthly wage will be → W = 0.07S + 5000
Given in the question,
Monthly salary of the sales-person = $5000Commission = 7% of total salesLet the total sales of a month = $S
Therefore, commission on this sale = 7% of $S
= $0.07S
Salary of the month = $5000
Total earnings (W) of the sales person = $(0.07S + 5000)
Therefore, equation for the monthly wage will be,
W = 0.07S + 5000
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A boat takes 3 days to travel from town A to town B, but it takes 4 days to travel from town B to town A. If a motor-less raft is left alone in the water by town A, how long will it take for the raft to float to town B?
Answers
Answer:
24 days
Step-by-step explanation:
The distance from A to B equals the distance from B to A.
Let the distance between A and B be d.
3 days = 72 hours
4 days = 96 hours
speed = distance/time
speeds are in miles per hour
speed from A to B = d/72
speed from B to A = d/96
difference in speeds:
d/72 - d/96 = d/288
The speed of the water is half of the difference.
speed = d/576
When the raft floats from A to B, it uses only the speed of the water.
d/576 / d/72 = 1/8
The speed of the water is 1/8 the overall speed of the trip from A to B, so traveling by the speed of the water alone must take 8 times longer than with the boat motor.
8 * 3 days = 24 days
A survey of 400 students yielded the following information: 262 were seniors, 215 were commuters, and 150 of the seniors were commuters. How many of the 400 surveyed students were seniors or were commuters?
Answers
Out of the 400 surveyed students, 327 were either seniors or commuters.
To find the number of students who were either seniors or commuters out of the 400 surveyed students, we need to add the number of seniors and the number of commuters while avoiding double-counting those who fall into both categories.
According to the information given:
There were 262 seniors.
There were 215 commuters.
150 of the seniors were also commuters.
To avoid double-counting, we need to subtract the number of seniors who were also commuters from the total count of seniors and commuters.
Seniors or commuters = Total seniors + Total commuters - Seniors who are also commuters
= 262 + 215 - 150
= 327
Therefore, out of the 400 surveyed students, 327 were either seniors or commuters.
It's important to note that in this calculation, we accounted for the overlap between seniors and commuters (150 students who were both seniors and commuters) to avoid counting them twice.
This ensures an accurate count of the students who fall into either category.
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What is the largestNumber of these wooden Els that can be packed in a box that is 2 cm x 4 cm x 6 cm
Answers
The largest number of the wooden Els with it's total surface area that can be packed in the 2cm×4cm×6cm box is 2 wooden Els.
Total Surface Area of Solid ShapesIn finding the total surface area of a solid cuboid, add the areas of all 6 faces. We can also label the length (l), width (w), and height (h) and use the formula, SA=2(lw+lh+hw), to find the surface area.
For the box, l=2cm, w=4cm and h=6cm
total surface area of box=2(2×6+2×4+6×4) cm square units
total surface area of box=2(44) cm square units
total surface area of box=88cm square units
For the top cuboid of the wooden El, l=3cm, w=1cm and h=2cm
total surface area of top El cuboid=22cm square units
For the bottom cuboid of the wooden El, l=1cm, w=1cm and h=2cm
total surface area of bottom El cuboid=10cm square units
total surface area of the El=32cm square units
(88cm²/32cm²)=2.75
This implies that only two(2) whole Els with total surface area of 32cm² can be packed in the box.
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Name one attribute of a square that is not an attribute
Answers
In ΔUVW, \overline{UW} UW is extended through point W to point X, \text{m}\angle UVW = (3x+16)^{\circ}m∠UVW=(3x+16) , \text{m}\angle WUV = (2x+8)^{\circ}m∠WUV=(2x+8) and \text{m}\angle VWX = (8x-18)^{\circ}m∠VWX=(8x−18) Find \text{m}\angle WUV.m∠WUV.
Answers
Applying the exterior angle of a triangle theorem, m<WUV = 36°
What is the Exterior Angle of a Triangle Theorem?The exterior angle of a triangle theorem states that the measure of the angle formed, when one side of a triangle is extended, is equal to the sum of the opposite interior angles of that triangle.
Applying the exterior angle of a triangle theorem:
m<WUV + m<UVW = m<VWX
Substitute
2x + 8 + 3x + 16 = 8x - 18
Add like terms
5x + 24 = 8x - 18
5x - 8x = -24 - 18
-3x = -42
-3x/-3 = -42/-3
x = 14
m<WUV = 2x + 8
Plug in the value of x
m<WUV = 2(14) + 8
m<WUV = 28 + 8
m<WUV = 36°
Therefore, applying the exterior angle of a triangle theorem, m<WUV = 36°
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FIND ALL THREE ANGLES
Answers
Answer:
Austin = 38
Atlanta = 89
Chicago = 89
Step-by-step explanation:
First, solve for x. All angles together equal 180. Set your formula up as
180 = 2x - 12 + 2x + 3 + 4x - 11
180 = 8x -20
180 + 20 = 8x
200 = 8x
200 / 8 = x
25 = x
Now substitute 25 in place of x
Austin
2x - 12
2*25-12 = 38
Atlanta
4x-11
4*25-11 = 89
Chicago
2x+3
2*25+3 = 53
Add all your angles together to check your work
(Bonus) A rectangular tank with a bottom and sides but no top is to have volume 500 cubic feet. Determine the dimensions (length, width, height) with the smallest possible surface area.
Answers
Answer:
Length and Width = 10ft
Height = 5ft
Surface Area = 300 square feet
Step-by-step explanation:
Given
\(V = 500ft^3\) -- Volume
Let:
\(L = Length\)
\(W =Width\)
\(H = Height\)
Volume (V) is calculated as:
\(V = L * W * H\)
Substitute 500 for V
\(500 = L * W * H\)
Make H the subject
\(H = \frac{500}{LW}\)
The tank has no top. So, the surface area (S) is:
\(S = L * W + 2*H*L + 2*H*W\)
\(S = L * W + 2H(L + W)\)
Substitute 500/LW for H
\(S = L * W + 2*\frac{500}{LW}(L + W)\)
\(S = L * W + \frac{1000}{LW}(L + W)\)
\(S = L W + \frac{1000}{L} + \frac{1000}{W}\)
Differentiate with respect to L and to W
\(S'(W) = L - \frac{1000}{W^2}\)
and
\(S'(L) = W - \frac{1000}{L^2}\)
Equate both to get the critical value
\(S'(W) = L - \frac{1000}{W^2}\)and \(S'(L) = W - \frac{1000}{L^2}\)
\(0 = L - \frac{1000}{W^2}\) and \(0 = W - \frac{1000}{L^2}\)
\(\frac{1000}{W^2} = L\) and \(\frac{1000}{L^2} = W\)
\(W^2L = 1000\) and \(L^2W = 1000\)
Make L the subject in \(W^2L = 1000\)
\(L = \frac{1000}{W^2}\)
Substitute \(\frac{1000}{W^2}\) for L in \(L^2W = 1000\)
\((\frac{1000}{W^2})^2 * W = 1000\)
\(\frac{1000000}{W^4} * W = 1000\)
\(\frac{1000000}{W^3} = 1000\)
Cross Multiply
\(1000000 = 1000W^3\)
Divide both sides by 1000
\(1000 = W^3\)
Take cube roots of both sides
\(\sqrt[3]{1000} = W\)
\(10 = W\)
\(W = 10\)
Substitute 10 for W in \(L = \frac{1000}{W^2}\)
\(L = \frac{1000}{10^2}\)
\(L = \frac{1000}{100}\)
\(L = 10\)
Recall that:\(H = \frac{500}{LW}\)
\(H = \frac{500}{10*10}\)
\(H = \frac{500}{100}\)
\(H = 5\)
So, the dimensions are:
\(L, W=10\) and \(H = 5\)
The surface area is:
\(S = L * W + 2H(L + W)\)
\(S = 10*10 +2*5(10+10)\)
\(S = 10*10 +2*5*20\)
\(S = 100 + 200\)
\(S = 300\)
Help if you know the answer
Answers
Answer:
k women women female mistake mistress milkmaid non mistress miss mama pottis potas potas
Simplify: √8^2-4×2×3
Answers
Answer:
-16
Step-by-step explanation:
8 - 4 x 2 x 3 =
8 - 24 =
-16 =
[64 - 4 x 2 x 3
Then we simplify 4 x 2 to 8
[64 - 8 x 3
Then we simplify 8 x 3 to 24
[64 - 24
Then we simplify 64 - 24 to 40
[40
we simplify again but this time [40 to 2{10
2[10
How many ¼ pound bags of nuts can be made from a 2 1/2 pound bag of nuts?
Answers
There are 16 ounces in a pound.
A 1/4 pound bag of nuts is equal to 1/4 * 16 = <<1/416=4>>4 ounces.
Therefore, you can make 40/4 = <<40/4=10>>10 1/4 pound bags of nuts from a 2 1/2 pound bag of nuts.
Answer:
9 bags
Step-by-step explanation:
2 1/4 ÷ 1/4
9/4 ÷ 1/4
9/4 x 4/1 = 9
Please help! Correct answer only, please! Consider the matrix shown below: What are the dimensions of A. A. 3 X 4 B. 4 X 3 C. 12 D. A and B
Answers
Answer: A) 3 x 4
Step-by-step explanation:
The dimensions of a matrix are ROWS x COLUMNS.
The given matrix has 3 rows and 4 columns,
therefore the dimensions are: 3 x 4
-12.405 as a mixed number in simplest form please
Answers
\(\\ \ast\sf\longmapsto -12.405\)
\(\\ \ast\sf\longmapsto -\dfrac{12405}{1000}\)
Simplify until possible\(\\ \ast\sf\longmapsto -\dfrac{2481}{200}\)
Now
\(\\ \ast\sf\longmapsto -12\dfrac{81}{200}\)
solve the following differential equation by variation of parameters. fully evaluate all integrals. find the most general solution to the associated homogeneous differential equation. use and in your answer to denote arbitrary constants, and enter them as c1 and c2. c1cos(4x) c2sin(4x) 1/16ln(cos(4x))cos(4x) 1/4xsin(4x) help (formulas) find a particular solution to the nonhomogeneous differential equation . help (formulas) find the most general solution to the original nonhomogeneous differential equation. use and in your answer to denote arbitrary constants. help (formulas)
Answers
The most general solution to the associated homogeneous differential equation is y=x/2-1/4
How will you solve this equation?C=0
dy/dx+2y=x
Use the formula:
\(\int\ \,xe^(2x)dx=e^(2x)\)((x/2−1/4).
We know that a linear differential equation is written in the standard form:
y' + a(x)y = f(x)
we get that: a(x)=2 and f(x)=x.
We know that the integrating factor is defined by the formula:
u(x)=\(e^{\int\ \, a(x) dx}\)
⇒ u(x)=\(e^{∫ 2 dx}\)= \(e^{2x}\)
The general solution of the differential equation is in the form:
y=\frac{ ∫ u(x) f(x) dx +C}{u(x)}
⇒ y=\frac{\(e^{2x}\)· x dx + 0}\({e^{2x}}\)
y=\frac{\(e^{2x}\) (x/2-1/4)\(}{e^{2x}\)
y=x/2-1/4
Hence, the most general solution to the associated homogeneous differential equation is y=x/2-1/4.
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Determine the slope of a line through (2,7) and (11,10)
Answers
Answer
Put in slope formula and you will have your answer
Step-by-step explanation:
hope this helps
helppppppppppppppppppppppppppppppppppppppppp
Answers
Answer:
y = -1/2x + 5/2
Step-by-step explanation:
the slope of the perpendicular line will be the opposite reciprocal of 2, which is -1/2
now substitute (7,-1) into 'x' and 'y' in slope-intercept equation of a line:
-1 = 7(-1/2) + b
-1 = -7/2 + b
b = 5/2
Given (x – 7)2 = 36, select the values of x. x = 13 x = 1 x = –29 x = 42
Answers
Answer:
x=25
Step-by-step explanation:
2x-14=36
2x=50
x=25
1 poinGiven the function tablebelow, what is the f(x)value when x = 4 (onlystate the numerical valuein your answer) *fix) = -5x + 3f(x)M-20Your answer
Answers
f(x)=-5x+3
Replace x by 4 and solve:
f(4) = -5(4)+3
f(4)= -20+3
f(4)= -17
Mr. Ahamad's science class is studying blood types. The table below shows the probability that a person living in the US has a particular blood type.
Answers
The probability that the three randomly selected students will have blood types A, B and AB is given as follows:
0.00164 = 0.164%.
How to calculate a probability?The parameters that are needed to calculate a probability are listed as follows:
Number of desired outcomes in the context of a problem or experiment.Number of total outcomes in the context of a problem or experiment.Then the probability is calculated as the division of the number of desired outcomes by the number of total outcomes.
Hence the probability for this problem is calculated as follows:
41/100 x 1/10 x 1/25 = 0.00164 = 0.164%.
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What is the range of the function f(x)=2x^3−3x for the domain {−1, 1, 2}?
Answers
The range of the function is {1,-1,10} for function f(x)=2x³−3x .
What is a function?A relation is a function if it has only One y-value for each x-value.
In a function what can go into a function is called the Domain. What may possibly come out of a function is called the Codomain. What actually comes out of a function is called the Range.
The given function is f(x)=2x³−3x
The domain is {-1,1,2}
Now f(-1)=2(-1)³-3(-1)
=-2+3=1
f(1)=2(1)³-3(1)=2-3=-1
f(2)=2(2)³-3(2)=16-6=10
Hence, the range of the function is {1,-1,10} for f(x)=2x³−3x .
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Whenever Deven and Laura owe each other money, they "pay" each other using stickers. They've agreed that a Harry Potter sticker is worth 49 dollars and a Twilight sticker is worth 35 dollars. They can even use stickers as "change" if one person overpays the other. For example, if Deven owes Laura 189 dollars, he can give her 6 Harry Potter stickers ($6 \cdot 49 = 294$ dollars), and she can return 3 Twilight stickers ($3 \cdot 35 = 105$ dollars). This trade is like a transfer of $294-105=189$ dollars. What is the smallest positive debt, in dollars, that can be paid off using sticker trading?
Answers
The smallest positive debt that can be paid off using sticker trading is $7$ dollars.
To find the smallest positive debt that can be paid off using sticker trading, we need to consider the values of the stickers (in dollars) and find the smallest positive amount that can be reached through a combination of these values.
Given that a Harry Potter sticker is worth $49 and a Twilight sticker is worth $35, we can approach this problem using the concept of the greatest common divisor (GCD) of these two values.
The GCD of $49$ and $35$ is $7$. This means that any multiple of the GCD can be represented using these sticker values.
In other words, any positive multiple of $7$ dollars can be paid off using sticker trading.
Therefore, the smallest positive debt that can be paid off using sticker trading is $7$ dollars.
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Simplify: (9^2 + 7^2)^3 + 10^3 pls help
Answers
Answer: 81 + 49 + 1000
Step-by-step explanation:
Answer: 2,198,000
Step-by-step explanation:
(81 + 49)^3 + 1000
2,197,000 + 1000
2198,000
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