[tex] \Large \bold{SOLUTION\ 1:} [/tex]
[tex] \small \begin{array}{l} \text{First, we need to check if the given differential} \\ \text{equation is exact.} \\ \\ (1-xy)^{-2} dx + \big[y^2 + x^2 (1-xy)^{-2}\big]dy = 0 \\ \\ \dfrac{dx}{(1-xy)^2} + \left[y^2 + \dfrac{x^2}{(1-xy)^2}\right]dy = 0 \\ \\ \quad M(x, y) dx + N(x, y) dy = 0 \end{array} [/tex]
[tex] \small \begin{array}{l l}\tt\: M(x,y) = \dfrac{1}{(1 - xy)^2}, & N(x,y) = y^2 + \dfrac{x^2}{(1-xy)^2}\\ \\\tt \dfrac{\partial M}{\partial y} = \dfrac{-2(-x)}{(1 - xy)^3}, & \dfrac{\partial N}{\partial x} = \dfrac{2x}{(1 - xy)^2} + \dfrac{-2(-y)x^2}{(1 - xy)^3} \\ \\\tt \dfrac{\partial M}{\partial y} = \dfrac{2x}{(1 - xy)^3}, & \dfrac{\partial N}{\partial x} = \dfrac{2x(1 - xy)+2x^2y}{(1 - xy)^3} \\ \\\tt \: & \dfrac{\partial N}{\partial x} = \dfrac{2x}{(1 - xy)^3} \end{array} [/tex]
[tex] \small \begin{array}{l} \tt\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x} \implies \text{Differential equation is exact.} \\ \\\tt \dfrac{\partial F}{\partial x} = M(x, y) = \dfrac{1}{(1 - xy)^2} \\ \tt\displaystyle F(x, y) = \int \dfrac{1}{(1 - xy)^2} \partial x = -\dfrac{1}{y} \int \dfrac{1}{(1 - xy)^2}(-y)\partial x \\ \\ \tt\:F(x, y) = \dfrac{1}{y(1 - xy)} + h(y) \\ \\ \tt\dfrac{\partial F}{\partial y} = N(x, y) = y^2 + \dfrac{x^2}{(1-xy)^2} \\ \\\tt \dfrac{\partial}{\partial y}\left[\dfrac{1}{y(1 - xy)} + h(y)\right] = y^2 + \dfrac{x^2}{(1-xy)^2} \\ \\ \tt-\dfrac{1 - xy + y(-x)}{y^2(1 - xy)^2} + h'(y) = y^2 + \dfrac{x^2}{(1-xy)^2} \\ \\ \tt-\dfrac{1 - 2xy}{y^2(1 - xy)^2} + h'(y) = y^2 + \dfrac{x^2}{(1-xy)^2} \\ \\ h'(y) = y^2 + \dfrac{x^2}{(1-xy)^2} + \dfrac{1 - 2xy}{y^2(1 - xy)^2} \\ \\ \tt\:h'(y) = y^2 + \dfrac{x^2y^2 - 2xy + 1}{y^2(1-xy)^2} = y^2 + \dfrac{1}{y^2} \\ \\ h(y) = \dfrac{y^3}{3} - \dfrac{1}{y} + C \\ \\ \tt\text{Substituting to }F(x,y),\text{we get} \\ \\ \dfrac{1}{y(1 - xy)} + \dfrac{y^3}{3} - \dfrac{1}{y} = C \\ \\ \quad \quad \text{or} \\ \\ \tt\red{\boxed{\dfrac{x}{1 - xy} + \dfrac{y^3}{3} = C} \Longleftarrow \textit{Answer}} \end{array} [/tex]
[tex] \Large \bold{SOLUTION\ 2:} [/tex]
[tex] \small \begin{array}{l} \tt\text{Since we already know that the equation is exact,} \\ \text{we can then continue solving for the solution by} \\ \text{inspection method or by algebraic manipulation.} \\ \\ \tt(1-xy)^{-2} dx + \big[y^2 + x^2 (1-xy)^{-2}\big]dy = 0 \\ \\ \tt\dfrac{dx}{(1-xy)^2} + \left[y^2 + \dfrac{x^2}{(1-xy)^2}\right]dy = 0 \\ \\ \tt\dfrac{dx}{(1-xy)^2} + y^2 dy + \dfrac{x^2}{(1-xy)^2} dy = 0 \\ \\ \tt\dfrac{dx + x^2dy}{(1-xy)^2} + y^2 dy = 0 \\ \\ \tt\text{Divide both numerator and denominator of the} \\ \tt\text{fraction by }x^2. \end{array} [/tex]
[tex] \small \begin{array}{c}\tt \dfrac{\dfrac{1}{x^2}dx + dy}{\dfrac{(1-xy)^2}{x^2}} + y^2 dy = 0 \\ \tt\\ \tt\dfrac{\dfrac{1}{x^2}dx + dy}{\left(\dfrac{1}{x}-y\right)^2} + y^2 dy = 0 \\ \\ \tt-\dfrac{\left(-\dfrac{1}{x^2}dx - dy\right)}{\left(\dfrac{1}{x}-y\right)^2} + y^2 dy = 0 \\ \\ \tt\displaystyle {\large{\int}} -\frac{d\left(\dfrac{1}{x}-y\right)}{\left(\dfrac{1}{x}-y\right)^2} + \int y^2 dy = \int 0 \\ \\ \tt\implies\tt \dfrac{1}{\dfrac{1}{x}-y} + \dfrac{y^3}{3} = C \\ \\\text{or} \\ \\ \tt\red{\boxed{\dfrac{x}{1 - xy} + \dfrac{y^3}{3} = C} \Longleftarrow \textit{Answer}} \end{array} [/tex]
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5h-2g=26
2h + 2g 2
find h and g
Answer:
g = -3, h = 4.
Step-by-step explanation:
5h - 2g = 26
2h + 2g = 2 Adding the 2 equations:
7h = 28
h = 4.
Substituting for h in the first equation:
5(4) - 2g = 26
-2g = 26 -20 = 6
g = -3.
find the zeros simplify all irrational and complex solutions
f(x)=x^4-11x^2-80
Answer:
Step-by-step explanation:
f(x)=x^4-11x^2-80=x^4-16x^2+5x^2-80
=x²(x²-16)+5(x²-16)
=(x²-16)(x²+5)
=(x-4)(x+4)[x²-(-5)]
=(x-4)(x+4)[x^2-(5i²)]
=(x-4)(x+4)(x-√5 i)(x+√5 i)
so zeros are 4,-4,√5 i,-√5 i
If EF || DG, and S and T are midpoints, then DG= ? Ef=13 ST=19
Answer: 25
Step-by-step explanation:
I’m stuck pls help me
Answer:
B and E
Step-by-step explanation:
Sally planted 3/8 rows of beans and 1/8 rows of Tyrion’s in a garden. In total how many rows of vegetables did sally plant?
Answer:
4/8 or 1/2 rows
Step-by-step explanation:
Hope this helps! ^^
How could you demonstrate that the force of gravity acting on an object is proportional to the mass of the object?
Answer:
F = mg
Step-by-step explanation:
g is the constant of proportionality and force is the product of mass and the acceleration due to gravity
f is directly proportional to m, therefore, f = mg
Help help help help help hel
Answer:
12
Step-by-step explanation:
P = 8
Q = 16
The value of PQ = 16 - 8 = 8
Since M is the midpoint of PQ,
PM = MQ
PM + MQ = PQ
PM + PM = PQ
2 PM = 8
PM = 8 / 2
PM = 4
The value of M
= 8 + 4
= 12
Answer:
M = 12
Step-by-step explanation:
If you want to find the midpoint between the numbers 8 and 16, follow the simple steps below.
First, add the two numbers: 8 + 16 = 24
Secondly, divide the total by 2: 24/2 = 12
The midpoint between 8 and 16 is 12.
hope this helps!! p.s. i really need brainliest :)
Donte is making brownies. For one batch, the recipe requires 1 5/6 cups of chocolate chips, and 1/6 cup of sugar. What is the combined amount, in cups, of chocolate chips and sugar that is used in one batch of brownies?
Answer:
2
Step-by-step explanation:
1 ⅚ + ⅙ = 1 6/6, also known as, 2
what is equivalent to 2
x
2
−
7
x
−
3
=
0
Answer:
I'm confused with this, I dont understand what you are asking.
Step-by-step explanation:
2 is equivalent with a lot of things,
1+1
3-1
92-90
7×7-47
4÷2
the list goes on...
A cuple travels 420 miles to dr0p their dughter off at summer camp. The trip back has a short cut and is only 360 miles. And the gas mileage is 50% better
due to a lighter load. Write an expression for the total gas used and simplify.
The expression for the simplified total gas used is 1.43x.
How to write the expression of the total gaslet the gas used on the way to camp "x"
Then the gas used on the way back can be calculated as follows:
360 / 420 × 0.5 × x
To calculate the total fuel used, We need to add this to the 'x' used on the way.
Therefore,
total gas used = x + 360 / 420 × 0.5 × x = x + 0.43x = 1.43x
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A test requires two hours to attend five questions if a candidate spent 80 minutes on the first two question what fraction of the time did he spend on the remaining three question
Answer:
He spent ⅓ of the time on the remaining three questions.
Step-by-step explanation:
Let's first figure out what the question gives us, and what we need to find.
Given:
Test - 2 hours, 5 questions
Candidate - 80 minutes, 2 questions
Find:
Fraction of time he spent on remaining three questions
Start by finding the fraction of time that he spent on the first two questions.
Set up an unit converter to convert two hours to minutes.
[tex]2$ hours($\frac{60\text{ minutes}}{1 \text{ hour}})=2\cdot 60\text{ minutes}=120$ minutes[/tex]
Let "x" represent the fraction of time the candidate spent on the first two questions...
[tex]x=\frac{\text{Amount of Time spent on First Two Questions}}{\text{Total Time}}\\x=\frac{80\text{ minutes}}{120\text{ minutes}}[/tex]
[tex]x=\frac{80}{120}[/tex]
[tex]x=\frac{8}{12}[/tex]
[tex]x=\frac{2}{3}[/tex]
Let "y" represent the fraction of time the candidate spent on the remaining three questions...
[tex]y=1-\text{Fraction of Time spent on First Two Questions}\\y=1-\frac{2}{3}\\[/tex]
[tex]y=\frac{3}{3}-\frac{2}{3}\\[/tex]
[tex]y=\frac{1}{3}[/tex]
He spent ⅓ of the time on the remaining three questions.
Complete the table below with the amounts of raspberry juice, apple
juice and soda needed to make the different quantities of Fruit Fizz.
mixture must taste exactly the same each time.
Given g(x)
-2 – 2, find g(-3).
Answer:
The calculation of the function given shows that since g(x) = -2x² - 9, the value of g(3) will be -27.
How to calculate the function?From the complete function, it should be noted that the value of g(x) = -2x² - 9.
Therefore, the value of g(3) will be calculated thus:
= -2(3)² - 9
= -2(9) - 9.
= -18 - 9
= -27
In conclusion, g(3) is -27.
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Look at this square. Find the value of t.
Answer:
t = 2
Step-by-step explanation:
The perimiter is the lenth of the out side of the square. Sense we know its a square we know that the side lenth are all the same and sense we have 4 sides we would have to devide 8 by 4
8/4
which equals
t=2
Answer:
Step-by-step explanation:
Perimeter of square:
Perimeter = 4*side
[tex]\boxed{Side =\dfrac{Perimeter}{4}}[/tex]
[tex]t=\dfrac{8}{4}[/tex]
t = 2 yards
Let w = width and 4 + w = length. Which equation represents the situation?
Answer:
Depends on whether you're looking for area or perimeter. Follow the equations below
Step-by-step explanation:
This depends on whether you're being asked to find perimeter of area
Perimeter is equal to 2(length) + 2(width)
In which case perimeter would be 2(4 + w) + 2(w)
Which can be simplified to 8 + 2w + 2w = 4w + 8
However, Area is equal to length times width
Which would be (4 + w) * w = 4w + w^2
Answer:
Step-by-step explanation:
✔ w(4 +w) = 60
What table represents the relationship “A video game designer makes $35 per hour”? A 2-column table with 3 rows. Column 1 is labeled hours with entries 35, 70, 105. Column 2 is labeled Dollars with entries 1, 2, 3. A 2-column table with 3 rows. Column 1 is labeled hours with entries 1, 2, 3. Column 2 is labeled Dollars with entries 35, 40, 45. A 2-column table with 3 rows. Column 1 is labeled hours with entries 1, 2, 3. Column 2 is labeled Dollars with entries 36, 37, 38. A 2-column table with 3 rows. Column 1 is labeled hours with entries 1, 2, 3. Column 2 is labeled Dollars with entries 35, 70, 105.
Answer:
d
Step-by-step explanation:
How much will the storage container
hold?
7 ft.
5 ft.
10 ft.
10 ft.
3 ft.
1 ft.
4 ft.
4 ft.
Answer:
The container will hold 190 ft^3 of volume of substance.
if the question means it.
This practice contains 15 questions. If you get 3 incorrect, what will your percent grade be? Hint: Grades are based on correct questions/total questions.
A. 30%
B. 0.8%
C. 20%
D. 80%
Answer:
D. 80%
Step-by-step explanation:
15-3 = 12
12/15 = 4/5 = .8
80%
Eric bought 8 pounds of flour for $5. How many pounds of flour did he get per dollar? If necessary, round your answer to the nearest hundredth.
Answer:
Step-by-step explanation:
8 pounds of flour = 5 dollars
1 pound of flour = x
Cross multiply
8x = 1*5 Divide by 8
8x/8 = 5/8
5/8 = 0.625
Therefore x = 0.625 dollars will buy 1 pound of flour.
Rounded x = 0.63
please help i'm asking to help the second time hope someone sees this I'll give 10 points and brainliest I promise
2. i) 1/2 × x
x/2
ii) 1/5 × x
x/5
iii)1-(1/2+1/5)
1- (5/10+2/10)
10/10- 7/10
3/10
3/10×x
3x/10
3. i)15× $p
15p
ii) cost for 4 bananas= $g
cost for 1 banana = $g÷4
cost for 16 bananas = $g/4 ×16
=$g×4
=4g
iii)
Cost for an orange= $p
Cost for 5 oranges = 5×$p
Cost for 1 banana= $g÷4
Cost for 2 bananas = 2× ($g÷4)
Add both of the expressions to get the total cost for 2 bananas and 5 oranges.
(2×$g÷4)+(5×$p)
($g×2/4)+(5×$p)
2/4 can be simplified as 1/2
g×1/2+5p
(g/2)+5p
Hope this helps!
One of the legs of a right triangle measures 2 cm and the other leg measures 17 cm. Find the measure of the hypotenuse. If necessary, round to the nearest tenth.
Answer:
17.1
Step-by-step explanation
a^2+b^2=c^2
2^2 + 17^2 = c^2
4 + 289 = c^2
293 = c^2
square root of 293 or 17.1
how many feet is the answer?
Answer:
1023 feet
Step-by-step explanation:
Sandy brought 6 1/4 dozen cookies to the bake sale. Alejandro brought 1 3/4 fewer cookies than Sandy brought. Pierre brought 2 3/4 dozen fewer cookies than Alejandro brought. How many dozen cookies did Pierre bring?
Answer: 4 1/2 doezen cookies
Step-by-step explanation: just subtract. Hope this helps.
When a constant force acts
upon an object, the acceleration of the object varies Inversely with Its mass. When a certain constant force acts upon an object witt
mass 5 kg, the acceleration of the object Is 15 m/s-
If the same force acts upon another object whose mass is 3 kg, what is this object's acceleration?
Answer:
25
Step-by-step explanation:
First expression: A= k/M
when A = 15 and M = 5, k = A×M = 75
therefore, when M = 4, A = 75/3 = 25
Find x on the triangle geometry please help
Answer:
[tex]\sf x=7\sqrt{6}[/tex]
Step-by-step explanation:
As the base angles of the triangle are the same, then the legs of the right triangle are also equal. Therefore, we can use Pythagoras' Theorem to calculate x.
Pythagoras' Theorem: a² + b² = c²
(where a and b are the legs, and c is the hypotenuse of the right triangle)
Given:
a = 7√3b = 7√3c = xSubstituting the given values into the formula:
[tex]\sf \implies (7\sqrt{3})^2+(7\sqrt{3})^2=x^2[/tex]
[tex]\sf \implies147+147=x^2[/tex]
[tex]\sf \implies x^2=294[/tex]
[tex]\sf \implies x=\sqrt{294}[/tex]
[tex]\sf \implies x=\sqrt{49 \cdot 6}[/tex]
[tex]\sf \implies x=\sqrt{49}\sqrt{6}[/tex]
[tex]\sf \implies x=7\sqrt{6}[/tex]
[tex]\\ \rm\Rrightarrow cos\theta=\dfrac{Base}{Hypotenuse}[/tex]
[tex]\\ \rm\Rrightarrow cos45=\dfrac{7\sqrt{3}}{x}[/tex]
[tex]\\ \rm\Rrightarrow \dfrac{1}{\sqrt{2}}=\dfrac{7\sqrt{3}}{x}[/tex]
[tex]\\ \rm\Rrightarrow x=7\sqrt{3}\sqrt{2}[/tex]
[tex]\\ \rm\Rrightarrow x=7\sqrt{6}[/tex]
Brand 1 Brand 2 Brand 3 Store 1 Store 2 Store 3 Laptop Comparison Chart Criteria for laptop search: Laptop specifications: • At least 3GB of memory • At least 250 GB hard drive • At least 14” Screen • Weigh at most 7 pounds • At least a 1 year manufacturer warranty • Built-in wireless router • Must be new, not used of refurbished Spending Limit: $800 • Includes laptop, sales tax, and shipping • For the purpose of this project, use 7.5% sales tax
The total amount that will be paid for the products with a sales tax of 7.5% will be $1290.
How to compute the amount from the sales tax?Your information isn't complete as the table denoting the appropriate values aren't given.
Let's assume that the price of a product A is given as $800 ane product B is $400. If the sales tax is 7.5%, the total amount that will be paid will be:
= ($800 + $400) + (7.5% × $1200)
= $1200 + $90
= $1290
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Gena and her friends each estimated the quotient of –137.56 divided by –6.12 using compatible numbers. Which shows the best estimate using compatible numbers?
Answer:
23
Step-by-step explanation:
Compatible numbers : numbers that are easy to add, subtract, multiply, or divide mentally. They are close in value to the actual numbers, although they do not necessary need to be rounded numbers.
-138 is close to -137.56
-6 is close to -6.12
[tex]\implies \dfrac{-138}{-6}[/tex]
Divide the top and bottom by -2:
[tex]\implies \dfrac{69}{3}[/tex]
Divide the top and bottom by 3
[tex]\implies 23[/tex]
If this is too difficult to divide mentally, you could use -140 and -7:
[tex]\implies \dfrac{-140}{-7}[/tex]
As 2 x 7 = 14, then 20 x 7 = 140, so the answer is 20.
Let y=6 and z=2. Translate “z less than y” into an algebraic expression and solve it
Answer:
y - z; 6 - 2; 4
Step-by-step explanation:
z less than y means subtract z from y
y - z
Substitute the given values.
6 - 2
4
2) Jose bought a magazine for $10 and some erasers for $2 per eraser. He spent a total of $26. How many erasers
he buy?
I can solve it i just need to know the equation with X
Please help! These data show the number of miles each of the 9 members of a track team ran for the week.
11 11.5 10.5 17 14.5 14.5 18 17 19
What is the IQR (interquartile range) for these data?
6.25
7.00
14.50
17.50
Answer:
14.50
Step-by-step explanation:
i took the quiz on k12 truse me i wish luck
The IQR for these data is 6.25.
Answer choice A is correct.
What is the interquartile range?An interquartile range is a measure of the difference between the upper and lower quartiles of a dataset.
W can find the values of the upper and lower quartile and median from a box plot.
The middle line is the median and the first line is the lower quartile and the last line is the upper quartile.
The formula used is Upper quartile - Lower quartile.
We have,
To find the IQR, we first need to find the first and third quartiles.
First, we need to order the data from smallest to largest:
10.5, 11, 11.5, 14.5, 14.5, 17, 17, 18, 19
Next, we find the median (middle value) of the data set.
Since we have an odd number of data points, the median is the middle number, which is 14.5.
To find the first quartile (Q1), we need to find the median of the lower half of the data set.
That is, we need to find the median of 10.5, 11, 11.5, and 14.5.
The median of this subset is (11 + 11.5) / 2 = 11.25.
To find the third quartile (Q3), we need to find the median of the upper half of the data set.
That is, we need to find the median of 17, 17, 18, and 19.
The median of this subset is (17 + 18) / 2 = 17.5.
Now we have the first quartile (Q1) = 11.25 and the third quartile (Q3) = 17.5. The IQR is the difference between Q3 and Q1:
IQR = Q3 - Q1
= 17.5 - 11.25
= 6.25
Therefore,
The IQR for these data is 6.25.
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