Suppose Sn is defined as 2 + 22 + 23 + . . . + 2n . What is the next step in your proof of Sn = 2(2n - 1), after you verify that Sn is valid for n = 1?
A. Show that Sn is valid for n = k + 2.
B. Assume that Sn is valid for n = k .
C. Verify that Sn is valid for n = 1.
D. Show that Sn is valid for n = k.

Answers

Answer 1

Remark:

[tex]S_n=2*1+2*2+2*3+...+2*n=2(1+2+3+...+n)[/tex]

[tex]1+2+3+...+n= \frac{n(n+1)}{2} [/tex], by the famous Gauss formula.

So the formula for [tex]S_n[/tex] is:

[tex]S_n=2*\frac{n(n+1)}{2}=n(n+1)[/tex]

these types of formulas are proven by Induction.

The first step is proving for n=1,

then the next step is assuming Sn is valid for n=k.

Answer: B. Assume that Sn is valid for n = k .

Related Questions

The tables represent two linear functions in a system.

What is the solution to this system?

Answers

The solution to this system is (x, y) = (8, -22).

The y-values get closer together by 2 units for each 2-unit increase in x. The difference at x=2 is 6, so we expect the difference in y-values to be zero when we increase x by 6 (from 2 to 8).

You can extend each table after the same pattern.

In table 1, x-values increase by 2 and y-values decrease by 8.

In table 2, x-values increase by 2 and y-values decrease by 6.

The attachment shows the tables extended to x=10. We note that the y-values are the same (-22) for x=8 (as we predicted above). That means the solution is ...

... (x, y) = (8, -22)

Answer:

(8,-22)

Step-by-step explanation:

Table 1)

To form equation we will use two point slope form

[tex](x_1,y_1)=(-4,26)\\(x_2,y_2)=(-2,18)[/tex]

Formula :[tex]y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]

Substitute the values :

[tex]y-26=\frac{18-26}{-2+4}(x+4)[/tex]

[tex]y-26=-4(x+4)[/tex]

[tex]y-26=-4x-16[/tex]

[tex]y=-4x-16+26[/tex]

[tex]y=-4x+10[/tex] ---1

Table 2)

To form equation we will use two point slope form

[tex](x_1,y_1)=(-4,14)\\(x_2,y_2)=(-2,8)[/tex]

Formula :[tex]y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]

Substitute the values :

[tex]y-14=\frac{8-14}{-2+4}(x+4)[/tex]

[tex]y-14=-3(x+4)[/tex]

[tex]y-14=-3x-12[/tex]

[tex]y=-3x+2[/tex] ---2

Now we are supposed to solve 1 and 2

Substitute the value of y from 1 in 2

[tex]-4x+10=-3x+2[/tex]

[tex]8=x[/tex]

Substitute the value of x in 2

[tex]y=-3(8)+2[/tex]

[tex]y=-22[/tex]

Hence the solution to this system is (8,-22)

Question help a robotic rover on mars finds a spherical rock with a diameter of 1313 centimeters. the rover picks up the rock and lifts it 1515 centimeters straight up. the rock has a specific gravity of 5.255.25. the gravitational acceleration on mars is 3.73.7 meters per second squared. if the robot's lifting arm has an efficiency of 3535% and required 1212 seconds to raise the rock 1515 centimeters, how much power (in watts) did the arm use?

Answers

Due to apparent doubling of numbers in the question, assume
diameter=13 cm/2=0.065m
h=15 cm=0.15m
rho=5.25*1000=5250 kg/m^3
eta=35%=0.35 [efficiency]
t=12 s
g=3.7 m/s^2

mass, m=rho*V=rho*(4pi/3)(r^3)
=5250 kg/m^3 * (4pi/3)(0.065)^3 m^3
=1.922 kg

Work done in lifting h=0.15m
W=mgh
=1.922pi kg * 3.7 m/s^2 * 0.15 m
=1.067pi kg (m/s)^2
=1.067pi J

Average power required, with efficiency eta=0.35
P=(W/t)/eta
=(1.067pi J )/ (12 s) /0.35
=0.254pi J/s
=0.798W (approx.)

You have been offered a project paying $300 at the beginning of each year for the next 20 years. what is the maximum amount of money you would invest in this project if you expect 9 percent rate of return to your investment?

Answers

This is a simple problem with some quick addition and subtraction, and a bit of multiplication.

First we figure out how much money you're spending in total, which is $300 each year, for 20 years, so 300 x 20 = 6,000. Now, since the problem is asking for a maximum amount for percentage rate, this means we find the 9% for all 20 years combined and not individually. 9% of 6,000 is of course 540, so if we subtract 540 from 6,000, we get 5,460.

This means your answer is $5,460

Breanna works in the mall and her hours increase during busy shopping times of the year. Breanna has a plan to save $25 each week for 5 weeks, and then increase that amount by $20 for the next 7 weeks during the busy time, then go back to $25 per week. If Breanna’s plan is to save $540, how many weeks total will she need to save?
a.
11 weeks
b.
16 weeks
c.
22 weeks
d.
23 weeks

Answers

The math I used is probably not the quickest way, but for me, the easiest.
25*5=125 25+20=45 45*7=315 125+315=440 25*4=100 440+100=540 5+7+4=16
B is your answer.

Answer:

Number of weeks Breanna saves are 16 weeks .

Option (b) is correct .

Step-by-step explanation:

As given

Breanna works in the mall and her hours increase during busy shopping times of the year.

Breanna has a plan to save $25 each week for 5 weeks.

Thus

Breanna saves in first 5 weeks = Amount saves per weeks in next 5 weeks × Number of weeks .

= $25 × 5

= $ 125

Increase that amount by $20 for the next 7 weeks during the busy time.

Thus

Amount saves per weeks for next 7 weeks = Amount saves per weeks in first 5 weeks + Increase in amount .

= $25 +$ 20

= $ 45

Breanna saves in next 7 weeks = Amount saves per week× Number of weeks .

Put all the values in the above

= $45 × 7

= $ 315

As given

Breanna saves $25 per week.

If Breanna’s plan is to save $540 .

Let us assume last weeks in which Breanna saves $25 per weeks be x .

Breanna saves in last x weeks = Amount saves for next x weeks × Number of weeks .

= 25 × x

= 25x

Equation becomes

Breanna saves in first 5 weeks + Breanna saves in next 7 weeks + Breanna saves in last x weeks = Total saving amount .

125 + 315 + 25x = 540

440 + 25x = 540

25x = 540 - 440

25x = 100

[tex]x = \frac{100}{25}[/tex]

x = 4 weeks

Thus

Total number of weeks Breanna saves = 5 weeks + 7 weeks + 4 weeks

= 16 weeks

Therefore the number of weeks Breanna saves are 16 weeks .

Option (b) is correct .

a soccar team spent $55 dollars on supplies for a car wash then earned $275 whats their total after paying for supplies

Answers

To find what profit they made, subtract what they earned from what they spent money on. 275-55=220. They made a total of $220 after paying for supplies. I hope this helps! :)

today there are 3,659 thousand elementary and secondary teachers employed in a certain country. this number is expected to increase to 3,856 thousand by next decade. what is the percent increase?

Answers

percent increase = (new number - original number) / original number...* 100
= (3856 - 3659) / 3659...* 100
= 197 / 3659....* 100
= 0.0538 * 100
= 5.38% <==

solve for y 6y-8=3y-2

Answers

6y-8 =3y-2

subtract 3y from both sides

3y-8 = -2

add 8 to both sides

3y=6

y=6/3 =2

y = 2

A store is selling two mixtures of coffee beans in one-pound bags. The first mixture has 12 ounces of Sumatra combined with 44 ounces of Celebes Kalossi, and costs $42. The second mixture has 44 ounces of Sumatra and 12ounces of Celebes Kalossi, and costs $30. How much does one ounce of Sumatra and one ounce of Celebes Kalossi cost?

Answers

Create 2 equations for the given problem

12s + 44c = 42
44s + 12c = 30

Take either equation and solve for either s or c.
12s + 44c = 42
12s = 42 - 44c
s = (42 - 44c) / 12
s = (21 - 22c) / 6

Substitute into 2nd equation

44s + 12c = 30
22((21 - 22c) / 3) + 12c = 30
22(21 - 22c) + 36c = 90
-448c +462 = 90
-448c = -372
c = -372/-448
c = 93/112
c = 0.83

Substitute into the 2nd equation

44s + 12c = 30
44s = 30 - 12c
s = (30 - 12 (93/112)) / 44
s = (30 - 279 / 28) / 44)
s = (30 - 9 27/28) / 44
s = 0.46

Prove
12s + 44c = 42
12(0.46) + 44(0.83) = 42
5.52 + 36.52 = 42
42.04 = 42

44s + 12c = 30
44(0.46) + 12(0.83) = 30
20.24 + 9.96 = 30.16

Assuming minor rounding issues this looks accurate.

A basket contains five apples and seven peaches. Four of the apples and two of the peaches are rotten. You randomly pick a piece of fruit. It is fresh or it is an apple. Find the probability of this occuring.

Answers

2/35possible outcomes:5*7=35fresh or apple = 22/35

Answer:[tex]\frac{5}{6}[/tex]

Step-by-step explanation:

Given

5 apples and 7 peaches out of which 4 apples and 2 peaches is rotten

i.e. 1 apple and 5 peaches is fresh

probability of fresh fruit is =[tex]\frac{6}{12}[/tex]

probability of apple =[tex]\frac{5}{12}[/tex]

Probability of both =[tex]\frac{1}{12}[/tex]

Probability of fresh or an apple is=[tex] \frac{6}{12}+\frac{5}{12}-\frac{1}{12}[/tex]

=[tex]\frac{5}{6}[/tex]

John has taken out a loan for college. He started paying off the loan with a first payment of $100. Each month he pays, he wants to pay back 1.1 times the amount he paid the month before. Explain to John how to represent his first 20 payments in sequence notation. Then explain how to find the sum of his first 20 payments, using complete sentences.

Answers

Let a_0 = 100, the first payment. Every subsequent payment is the prior payment, times 1.1. In order to represent that, let a_n be the term in question. The term before it is a_n-1. So a_n = 1.1 * a_n-1. This means that a_19 = 1.1*a_18, a_18 = 1.1*a_17, etc. To find the sum of your first 20 payments, this sum is equal to a_0+a_1+a_2+...+a_19. a_1 = 1.1*a_0, so a_2 = 1.1*(1.1*a_0) = (1.1)^2 * a_0, a_3 = 1.1*a_2 = (1.1)^3*a_3, and so on. So the sum can be reduced to S = a_0 * (1+ 1.1 + 1.1^2 + 1.1^3 + ... + 1.1^19) which is approximately $5727.50

To make the two triangles below similar, what would the values of x and y have to be?

Answers

i dont really remember how to solve this since i done this some time ago the angles must be the same i would say d since 5 the bigger triangle is five times bigger

~ hope this helps ~

First, y and 40 degrees would have to be the same; thus, y = 40 degrees.

Second, form and solve the ratio (x/25) = (6/30). 30x=25(6)=150. Thus, x=5.

Three times a number added to seven is 22

Answers

3x+7=22

3x=15

x=5

the number is 5

A pizza parlor sold 38 1/8 pizzas during a dinner hour. If each pizza contained 8 slices, how many slices of pizza were sold?

Answers

38*8 = 304

304 + 1 =305 slices were sold

8(38 1/8)

8(38)+8(1/8)

304+1

305 slices.

you have 5 antiques and want to display 3 on a shelf. how many different 3 antique arrangements are possible

Answers

[tex]\bf _{n}P_{r}=\cfrac{n!}{(n-r)!}\qquad\qquad _{5}P_{3}=\cfrac{5!}{(5-3)!}[/tex]

Determine whether the given differential equation is exact. if it is exact, solve it. (if it is not exact, enter not.) (3x + 6y) dx + (6x − 8y3) dy = 0

Answers

[tex]\underbrace{(3x+6y)}_{M(x,y)}\,\mathrm dx+\underbrace{(6x-8y^3)}_{N(x,y)}\,\mathrm dy=0[/tex]

The ODE is exact if [tex]\dfrac{\partial M}{\partial y}=\dfrac{\partial N}{\partial x}[/tex]. We have

[tex]\dfrac{\partial M}{\partial y}=6[/tex]
[tex]\dfrac{\partial N}{\partial x}=6[/tex]

so the equation is indeed exact.

We're looking for a solution of the form

[tex]\Psi(x,y)=C[/tex]

By the chain rule, we have

[tex]\dfrac{\partial\Psi}{\partial x}+\dfrac{\partial\Psi}{\partial y}\dfrac{\mathrm dy}{\mathrm dx}=0[/tex]
[tex]\iff\dfrac{\partial\Psi}{\partial x}\,\mathrm dx+\dfrac{\partial\Psi}{\partial y}\,\mathrm dy=0[/tex]

which means [tex]\dfrac{\partial\Psi}{\partial x}=M[/tex] and [tex]\dfrac{\partial\Psi}{\partial y}=N[/tex].

Now

[tex]\dfrac{\partial\Psi}{\partial x}=M\implies\displaystyle\int\frac{\partial\Psi}{\partial x}\,\mathrm dx=\int M\,\mathrm dx[/tex]
[tex]\implies\Psi(x,y)=\dfrac32x^2+6xy+f(y)[/tex]

Differentiating with respect to [tex]y[/tex] yields

[tex]\dfrac{\partial\Psi}{\partial y}=N\implies 6x+f'(y)=6x-8y^3[/tex]
[tex]\implies f'(y)=-8y^3\implies f(y)=\displaystyle\int(-8y^3)\,\mathrm dy=-2y^4+C[/tex]

and so the general solution is

[tex]\Psi(x,y)=\dfrac32x^2+6xy-2y^4+C=C[/tex]
[tex]\implies\Psi(x,y)=\dfrac32x^2+6xy-2y^4=C[/tex]

Final answer:

The given differential equation (3x + 6y) dx + (6x − 8y3) dy = 0 is exact because its mixed partial derivatives are equal. To solve it, first integrate M with respect to x to get a function involving an unknown function of y, then compare it with N to determine the unknown function.

Explanation:

To determine if the given differential equation is exact, we need to check if the partial derivative of M with respect to y is the same as the partial derivative of N with respect to x. In this given differential equation (3x + 6y) dx + (6x − 8y3) dy = 0, M = 3x + 6y and N = 6x - 8y3. First, compute the partial derivative of M with respect to y (∂M/∂y) which is 6, then the partial derivative of N with respect to x (∂N/∂x), which is 6 as well. Since ∂M/∂y=∂N/∂x, the given differential equation is exact.

To solve this exact differential equation, we integrate M dx, i.e., ∫M dx, to get ψ(x,y) =1.5x^2+6xy+h(y), where h(y) is an arbitrary function of y. Next, differentiate ψ(x,y) with respect to y, then compare the result with N: ψy=6x+dh/dy=6x-8y³. Solving for dh/dy gives dh/dy=-8y³, so integrating this gives h(y)=-2y⁴+C, where C is a constant. Therefore, the solution to the differential equation is ψ(x,y)=1.5x²+6xy-2y⁴=C.

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Each of a certain type of tire on a 4 wheel car lasts exactly 10,000 miles. How many spare tires of this same variety must carried so that a trip of 15,000 miles can just be completed.

Answers

3 spare tires are needed. This works if you rotate the tires every 5,000 miles.

Tire A, B, C, D (and spare is E F G)
first 5,000 miles you use tire A B C D
second 5000 miles you use tire B C D E
third 5000 miles you use A E F G

Each tires is used no more than twice (10,000 miles).

If 5 workers can pave 10 driveways in 30 days, how many days would it take 20 workers to pave 38 driveways?

Answers

Given that 5 workers can pave 10 driveways in 30 days, then:
let x be the number of days that can take 20 workers to complete 38 drive ways;
therefore:
proportion of work completed by 5 workers will be:
5*30/10
=15
proportion of work completed by 20 workers will be:
20*x/38
The time taken for 20 workers to complete 38 driveways will be:
15=20/38x
x=15*38/20
x=28 1/2
The answer is 28.5 days

Use rounding or compatible numbers to estimate sum 198+727

Answers

198 round to 200
727 round to 730
200+ 730 is 930

The answer is 930. Hope this helps!

palace help me solve these two equations 14, 15

Answers

14)

check the picture below... so.. is just a semi-circle and a rectangle, nothing fancy, get their area, sum them up.

15)

so... the voter turnout, is referring to, what's the percentage that voted, which year has the highest percentage... well, we simply check the percentage of the value provided from the total eligible voters, keeping in mind that, all eligible voters are the 100%

so.. for 1944

[tex]\bf \begin{array}{ccllll} amount&\%\\ ------&----\\ 86,607,000&100\\ 47,976,670&x\\ \end{array}\implies \cfrac{86,607,000}{47,976,670}=\cfrac{100}{x} \\\\\\ x=\cfrac{100\cdot 47,976,670}{86,607,000}\implies x\approx 55.39583405498[/tex]

for 1984

[tex]\bf \begin{array}{ccllll} amount&\%\\ -----&-----\\ 167,727,000&100\\ 92,659,000&x \end{array}\implies \cfrac{167,727,000}{92,659,000}=\cfrac{100}{x} \\\\\\ x=\cfrac{100\cdot 92,659,000}{167,727,000}\implies x\approx 55.2439380660239[/tex]

486 is what percent of 900

Answers

486/900 = .54 = 54%. .......

It is desired to estimate the mean gpa of each undergraduate class at a large university. assume that the variance of the gpas is 1.44. how large a sample is necessary to estimate the mean gpa within 0.25 at the 99% confidence level

Answers

Final answer:

To estimate the mean GPA of each undergraduate class at a large university within a margin of error of 0.25 at a 99% confidence level, approximately 187 samples are needed.

Explanation:

To estimate the mean GPA of each undergraduate class at a large university within a margin of error of 0.25 at a 99% confidence level, we need to determine the required sample size. In this case, we can use the formula for sample size determination, which is:

n = (Z^2 * σ^2) / E^2

Where:

n is the required sample sizeZ is the Z-score associated with the desired confidence level (99% corresponds to a Z-score of approximately 2.58)σ^2 is the variance of the GPAs (given as 1.44)E is the desired margin of error (0.25)

Plugging in the values, we get:

n = (2.58^2 * 1.44) / 0.25^2

n ≈ 186.43

Therefore, we need a sample size of approximately 187 to estimate the mean GPA within a margin of error of 0.25 at a 99% confidence level.

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What are the X intercepts of a parabola with vertex (6,27) and y intercept of (0,-81)?

Answers

so hmm let's see if we can get the parabola's equation

if we assume a vertical parabola, "y" in x-terms.... then

[tex]\bf \begin{array}{llll} \boxed{y=a(x-{{ h}})^2+{{ k}}}\\\\ x=a(y-{{ k}})^2+{{ h}} \end{array} \qquad\qquad vertex\ ({{ h}},{{ k}})\\\\ -------------------------------\\\\[/tex]

[tex]\bf \textit{we know that } \begin{cases} h=6\\ k=27 \end{cases}\implies y=a(x-6)^2+27 \\\\\\ \textit{we also know the y-intercept } \begin{cases} x=-81\\ y=0 \end{cases}\implies 0=a(-81-6)^2+27 \\\\\\ -27=a(-87)^2\implies \cfrac{-27}{(-87)^2}=a\implies -\cfrac{3}{841}=a\\\\ -------------------------------\\\\ \boxed{y=-\cfrac{3}{841}(x-6)^2+27}[/tex]

now, to get the x-intercepts, we simply set y = 0

[tex]\bf 0=-\cfrac{3}{841}(x-6)^2+27\implies -27=-\cfrac{3}{841}(x-6)^2 \\\\\\ \cfrac{-27\cdot 841}{-3}=(x-6)^2\implies 7569=(x-6)^2\implies \pm\sqrt{7569}=x-6 \\\\\\ \pm 87=x-6\implies \pm 87+6=x\implies x= \begin{cases} 93\\ -81 \end{cases}[/tex]

and... since the y = 0, then (93, 0) and (-81, 0)

what is the area of a circle with a radius of 7 inches

Answers

Area formula is A=πr2
A=3.1416(radius)2 (radius is 7)
A=3.1416(49)
A=153.94
The area is 153.94 inches.
Hope that was helpful :)

Answer is provided in the image attached.

The social studies teacher is giving a test worth 80 points, and containing 34 questions. There are 2 point and 4 point questions on the test. How many of each type of question are on the test? Show your work. A. List the two equations B. Find each type of question

Answers

x is 2 pt questions
y is 4 pt questions
x+y=34
2x+4y=80
Multiply the top equation by -2 to use elimination technique
-2x-2y=-68
Then combine the two equations (x is cancelled out)
2y=12
y=6
If there are 6 four point questions, there must be 28 two point questions to get a total of 34 questions.

Help Please question in the file.

Answers

check the picture below.

Sam is observing the velocity of a car at different times. After two hours, the velocity of the car is 50 km/h. After six hours, the velocity of the car is 54 km/h.

Part A: Write an equation in two variables in the standard form that can be used to describe the velocity of the car at different times. Show your work and define the variables used. (5 points)

Part B: How can you graph the equation obtained in Part A for the first seven hours? (5 points)

Answers

part A)

[tex]\bf \begin{array}{ccllll} hours(x)&velocity(y)\\ -----&-----\\ 2&50\\ 6&54 \end{array}\\\\ -------------------------------\\\\ \begin{array}{lllll} &x_1&y_1&x_2&y_2\\ % (a,b) &({{ 2}}\quad ,&{{ 50}})\quad % (c,d) &({{ 6}}\quad ,&{{ 54}}) \end{array}[/tex]

[tex]\bf slope = {{ m}}= \cfrac{rise}{run} \implies \cfrac{{{ y_2}}-{{ y_1}}}{{{ x_2}}-{{ x_1}}}\implies \cfrac{54-50}{6-2}\implies \cfrac{4}{4}\implies 1 \\\\\\ % point-slope intercept y-{{ y_1}}={{ m}}(x-{{ x_1}})\implies y-50=1(x-2)\\ \left. \qquad \right. \uparrow\\ \textit{point-slope form} \\\\\\ y-50=x-2\implies \boxed{y-x=48}\impliedby \begin{array}{llll} standard\\ form \end{array}[/tex]

part B)

well, to graph a LINEar equation, since it's just a LINE, you simply need two points to graph a line, and since we know that y - x = 48.... if you want to know what's "y" when x = 7? or 7 hours, well

y - (7) = 48
y = 48 + 7
y = 55

so.. that's the point for the 7th hour, 7, 55, and you can pick any other point above, and graph it away.

Given a positive integer n, assign true to is_prime if n has no factors other than 1 and itself. (remember, m is a factor of n if m divides n evenly.)

Answers

This is the following condition in order to get the specific output for this specific problem: if is_a_prime(n): is_prime = True Now all you have to do is write is_a_prime().

For the hard code for this problem:

if n == 2:
is_prime = True
elif n % 2 == 0:
is_prime = False
else:
is_prime = True
for m in range (3, int (n * 0.5) + 1, 2):
if n % m == 0:
is_prime = False
break.

To add, a high-level programming language that is widely used for general-purpose programming, created by Guido van Rossum and first released in 1991 is called Python.

Below is the solution of assign true to is_prime if n has no factors other than 1 and itself.

Further explanation

Python is an interpreted, high-level, general-purpose programming language. Python is created by Guido van Rossum and first released in 1991. Its design philosophy emphasizes code readability. Python is a more productive language than Java because it is an interpretive language which is accompanied by elegant syntax and it makes an excellent choice for scripting and rapid application development in many areas. Python is a general-purpose, versatile and popular because it is concise and easy to read, and it is also a good language to have in any programmer’s stack as it can be used for everything such as web development, software development, data science applications.

Given a positive integer n, assign true to is_prime if n has no factors other than 1 and itself. And remember, m is a factor of n if m divides n evenly. So:

if n == 2:

is_prime = True

elif n % 2 == 0:

is_prime = False

else:

is_prime = True

for m in range (3, int(n**0.5)+1, 2):

if n % m == 0:

is_prime = False

Learn moreLearn more about python https://brainly.com/question/4331067Learn more about python IDLE or similar environment https://brainly.com/question/6280029Learn more about python function https://brainly.com/question/9806744

Answer details

Grade: 9

Subject: mathematics

Chapter: python

Keywords: python, integer, factors, python IDLE, is_prime

py+7=6y+q
what is the answer for this question (linear equations with unknown coefficients)?

Answers

Assuming your solving for y:
py+7=6y+q
-6y -7 -6y -7
(p-6)y = q-7
divide both sides by p-6
y=(q-7)/(p-6)

Answer:

The required equation is [tex]y =\frac{(q-7)}{(p-6)}[/tex].

Step-by-step explanation:

Consider the provided equation.

[tex]py+7=6y+q[/tex]

We need to simplify the equation and convert it into a linear equation with unknown coefficients.

Subtract 6y from both sides.

[tex]py - 6y + 7 = q[/tex]

Subtract 7 from both sides.

[tex]py - 6y = q - 7[/tex]

Take y common from left side.

[tex]y(p-6) = q-7[/tex]

Divide both sides by p-6.

[tex]y =\frac{(q-7)}{(p-6)}[/tex]

Hence, the required equation is [tex]y =\frac{(q-7)}{(p-6)}[/tex].

Describe the relationship between the similarity ratio of two triangles and the ratio of their areas.

Answers

[tex]\bf \textit{let's say the sides ratio is }\cfrac{\bigtriangleup_1}{\bigtriangleup_2}\qquad \cfrac{s_1}{s_2} \\\\\\ \textit{then the ratio of their areas is }\cfrac{\bigtriangleup_1}{\bigtriangleup_2}\qquad \cfrac{(s_1)^2}{(s_2)^2}\\\\ -------------------------------\\\\[/tex]

[tex]\bf \qquad \qquad \textit{ratio relations} \\\\ \begin{array}{ccccllll} &Sides&Area&Volume\\ &-----&-----&-----\\ \cfrac{\textit{similar shape}}{\textit{similar shape}}&\cfrac{s}{s}&\cfrac{s^2}{s^2}&\cfrac{s^3}{s^3} \end{array} \\\\ -----------------------------\\\\ \cfrac{\textit{similar shape}}{\textit{similar shape}}\qquad \cfrac{s}{s}=\cfrac{\sqrt{s^2}}{\sqrt{s^2}}=\cfrac{\sqrt[3]{s^3}}{\sqrt[3]{s^3}}[/tex]

Find all solutions to the equation csc theta +sqrt2 = 0

Answers

Theta = 7pi/4+2pi*n
Theta = 5pi/4+2pi*n

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