Linear equations linear equations

The quadrilateral PQRS is a parallelogram since opposite sides PQ and SR have the same lengths and slopes.

To determine the correctness of the statements and calculate the lengths and slopes of the given quadrilateral PQRS, let's perform a coordinate proof.

Length and slope of PQ:

The coordinates of points P and Q are P(0, 2) and Q(3, -4), respectively.

Length of PQ: √[(x₂ - x₁)² + (y₂ - y₁)²]

= √[(3 - 0)² + (-4 - 2)²]

= √[9 + 36]

= √45

= 3√5 (approx.)

Slope of PQ: (y₂ - y₁) / (x₂ - x₁)

= (-4 - 2) / (3 - 0)

= -6 / 3

= -2

Therefore, the length of PQ is approximately 3√5, and its slope is -2.

Length and slope of SR:

The coordinates of points S and R are S(-2, 1) and R(1, -5), respectively.

Length of SR: √[(x₂ - x₁)² + (y₂ - y₁)²]

= √[(1 - (-2))² + (-5 - 1)²]

= √[9 + 36]

= √45

= 3√5 (approx.)

Slope of SR: (y₂ - y₁) / (x₂ - x₁)

= (-5 - 1) / (1 - (-2))

= -6 / 3

= -2

Therefore, the length of SR is approximately 3√5, and its slope is -2.

Length and slope of SP:

The coordinates of points S and P are S(-2, 1) and P(0, 2), respectively.

Length of SP: √[(x₂ - x₁)² + (y₂ - y₁)²]

= √[(0 - (-2))² + (2 - 1)²]

= √[4 + 1]

= √5

Slope of SP: (y₂ - y₁) / (x₂ - x₁)

= (2 - 1) / (0 - (-2))

= 1 / 2

Therefore, the length of SP is √5, and its slope is 1/2.

Length and slope of HQ:

The coordinates of points H and Q are H(0, 2) and Q(3, -4), respectively.

Length of HQ: √[(x₂ - x₁)² + (y₂ - y₁)²]

= √[(3 - 0)² + (-4 - 2)²]

= √[9 + 36]

= √45

= 3√5 (approx.)

Slope of HQ: (y₂ - y₁) / (x₂ - x₁)

= (-4 - 2) / (3 - 0)

= -6 / 3

= -2

Therefore, the length of HQ is approximately 3√5, and its slope is -2.

Based on the calculations, we can conclude that both statements are correct:

The quadrilateral PQRS is a parallelogram since opposite sides PQ and SR have the same lengths and slopes.

The quadrilateral PQRS is not a rectangle since the lengths of the adjacent sides PQ and SP are not equal.

Summary:

Length of PQ is approximately 3√5, and its slope is -2.

Length of SR is approximately 3√5, and its slope is -2.

Length of SP is √5, and its slope is 1/2.

Length of HQ is approximately 3√5, and its slope is -2.

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Answer: 18.78 cubic feet

Step-by-step explanation:

The detailed analysis is attached below.

Step-by-step explanation:

We can begin by using the fact that the selling price was 117.81, including a sales tax of 5%. Let x be the original price of the lawnmower.

After the discount of 32%, the selling price is:

0.68x + 0.05(0.68x) = 117.81

Simplifying and solving for x, we get:

0.68x + 0.034x = 117.81

0.714x = 117.81

x = 165

Therefore, the original price of the lawnmower was $165.

The first five terms of the following generalized Fibonacci sequence are -19, 14, -5, 9, 4

From the question, we have the following parameters that can be used in our computation:

The generalized Fibonacci sequence

In the sequence, we can see that the last two terms are added to get the new term

Also, we have

a(1) = -19

a(2) = 14

Using the above as a guide, we have the following:

a(3) = -19 + 14 = -5

a(4) = -5 + 14 = 9

a(5) = 9 - 5 = 4

Hence, the first five terms of the following generalized Fibonacci sequence are -19, 14, -5, 9, 4

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The balance after 4 years on $2000 at 4% is equal to $2320.

In Mathematics, simple interest can be calculated by using this formula:

S.I = PRT or S.I = A - P

Where:

Substituting the given parameters into the simple interest formula, we have;

S.I = 2000 × 4/100 × 4

S.I = 2000 × 0.04 × 4

S.I = $320.

Next, we would calculate the future value as follows;

Future value, A = S.I + P

Future value, A = $320 + $2000

Future value, A = $2320.

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The solution for x, is x= (3m - b)/2. (Option-d)

To solve for x in the given equation, m=(2x+b)/3, we can start by isolating x on one side of the equation.

First, we can multiply both sides of the equation by 3 to get rid of the denominator:

3m = 2x + b

Next, we can subtract b from both sides of the equation:

3m - b = 2x

Finally, we can divide both sides of the equation by 2:

x = (3m - b)/2

It's important to note that this is a linear equation with one unknown variable, and we can use algebraic methods to solve for the unknown variable. This equation may arise in many fields of study, including physics, engineering, and economics, among others.(Option-d)

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Answer:

42

Step-by-step explanation:

Rectangle: A = 6 x 5 = 30

Triangle: A = 1/2(6 x 4) = 12

Area of figure: 30 + 12 = 42

Answer:

area of rectangle+area of triangle= area of figure

6×5 + 1/2×4×6=30+24= 42

or

given figure is a trapezium

so we know are of trapezium= 1/2×sum of parallel sides×height

= 1/2×( 5+9)×6

=42

Given statement solution is :- On average, she spent approximately $115 per visit.

To calculate the total amount spent on the visits, we need to add up the co-payments and the cost of the shots.

Co-payments:

The co-pay for each visit is $25, and you visited the doctor 3 times a week for 8 weeks within 2 months. So the total co-payment is:

3 visits/week * $25/visit * 8 weeks = $600.

Shots:

You received the shot 2 times out of each visit, and you visited the doctor 3 times a week for 8 weeks within 2 months. So the total cost of the shots is:

2 shots/visit * $135/shot * 8 weeks = $2,160.

Total amount spent:

Co-payments + Shots = $600 + $2,160 = $2,760.

Therefore, the total amount spent for the visits and shots over 2 months is $2,760.

Trivia Answer:

To find out the average amount spent per visit, we can divide the total amount spent by the total number of visits. In this case, the total amount spent is $2,760, and the total number of visits is 3 visits/week * 8 weeks = 24 visits.

Average spent per visit = Total amount spent / Total number of visits

Average spent per visit = $2,760 / 24 visits

Average spent per visit ≈ $115.

Therefore, on average, you spent approximately $115 per visit.

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Answer: 4 small bags, 3 large bags (equations below in explanation)

Step-by-step explanation:

x=number of small bags

y=number of large bags

x+y=7

4x+10y=52

Subtract them after multiplying the first equation by 4.

4x+4y=28

4x+10y=52

-6x=-24

x=4

x+y=7

4+y=7

y=3

The length of each edge of the cube =  6 inches, Thus option B is correct.

The volume of a cube = 216 cubic inches,

length of each edge of a cube is an inch

To find the value of a:

The formula for the volume of a cube is [tex]a^{3}[/tex]

The volume of cubic = [tex]a^{3}[/tex]

V = [tex]a^{3}[/tex]

substitute the volume value given in the equation in the above formula

216 = [tex]a^{3}[/tex]

[tex]a = \sqrt[3]{216}[/tex]

[tex]a[/tex] = 6

a = 6inches

Therefore, the length of each edge of the cube is 6 inches.

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Answer:

Step-by-step explanation:

i am working on the assumption that nobody does all three of them

i got 4 because including the people that do swimming and park, the total number of people that do swimming is 22.

the same logic goes for cycling: including the people that do swimming and visit the national park, the total is 23.

so that means that find how many people do swimming and cycling, we have to add the people doing only swimming, with the people doing both swimming and park and then subtract that answer from 22 which gives you 4

The product of the expression 7 x 63 is 441.

We have,

To find the product of 7 x 63 using the distributive property, we can break down 63 as the sum of its factors, such as 60 and 3:

7 x 63 = 7 x (60 + 3)

Now, we can apply the distributive property by multiplying 7 to each term inside the parentheses:

7 x (60 + 3) = 7 x 60 + 7 x 3

Simplifying further:

7 x 60 + 7 x 3 = 420 + 21

Therefore,

The product of 7 x 63 is 441.

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Answer:160 students

Step-by-step explanation:

did I do it

(your answer needs to be more than 20 characters long)

To calculate the amount of trace element collected by Robert, we need to find the volume of the cylinder-shaped container. Rounded to the nearest tenth, Robert has collected approximately 211.4 grams of the trace element in the second sample.

To calculate the amount of trace element collected by Robert, we need to find the volume of the cylinder-shaped container and then multiply it by the concentration of the trace element.

The volume of a cylinder is given by the formula:

V = π * r^2 * h

where V is the volume, π is approximately 3.14159, r is the radius, and h is the height.

Given:

Radius (r) = 8.7 cm

Height (h) = 17.3 cm

Concentration of trace element = 0.052 grams/cm³

Substituting these values into the volume formula:

V = 3.14159 * (8.7 cm)^2 * 17.3 cm

V ≈ 4068.57196 cm³

Now, we can calculate the amount of trace element collected by multiplying the volume by the concentration:

Amount of trace element = V * concentration

Amount of trace element ≈ 4068.57196 cm³ * 0.052 grams/cm³

Amount of trace element ≈ 211.42941792 grams

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[tex]\qquad \textit{Amount for Exponential Decay} \\\\ A=P(1 - r)^t\qquad \begin{cases} A=\textit{current amount}\\ P=\textit{initial amount}\dotfill &5918\\ r=rate\to 6\%\to \frac{6}{100}\dotfill &0.06\\ t=years\dotfill &8\\ \end{cases} \\\\\\ A = 5918(1 - 0.06)^{8} \implies A = 5918( 0.94 )^{8}\implies A \approx 3607.43[/tex]

Answer:

Approx 8.47 [tex]m^2[/tex]

Step-by-step explanation:

First, find the area of the entire figure. Which can be found by multiplying the two sides of the square by each other=  [tex]6*6=36m^2[/tex]

Now, ignore the square and just focus on the pentagon and finding the area of that. The formula for the area of a pentagon is [tex]A=1/2Apothem*Perimeter[/tex]

In order to find the apothem, you can make a mini triangle and use trig (Tangent) to find the height (apothem) of the triangle. Then, you take the side length of the base of the pentagon which is 4 and then you multiply it by the number of sides (5) to get 20. Then, by plugging it into our formula, you use the apothem*10 (divided by 2) and you get the area, and then you just subtract that from 36.

The  expression represents the total surface area of the prism is

2 (5 * 7) + 2 (4 * 7) +  2  (4 * 5)

The term "TSA" stands for "Total Surface Area" of a prism. The Total Surface Area represents the  sum of the areas of all the faces (including the bases) of the prism.

for the rectangular prism, the Total Surface Area can be calculated using the formula:

TSA = 2lw + 2lh + 2wh

where

l = 5

w = 4

h = 7

plugging in the values gives

TSA = 2 (5 * 7) + 2 (4 * 7) +  2  (4 * 5)

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The complete table for the function are

- f(1) - 3 = -5

- f(0) - 3 = -5.5

- f(2) - 3  = -7

- f(6) - 3 = undefined

- f(14) - 3 = -1

- f3(0) - 3 = 2

From the question, we have the following parameters that can be used in our computation:

The function equation and the incomplete table of values

This is given as

g(x) = - f(x) - 3

From the table, the missing values are at all values of x

So, we have

- f(1) - 3 = -2 - 3 = -5

- f(0) - 3 = -2.5 - 3 = -5.5

- f(2) - 3 = -4 - 3 = -7

- f(6) - 3 = undefined

- f(14) - 3 = 2 - 3 = -1

- f3(0) - 3 = 5 - 3 = 2

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An equation for the total cars and trucks for dealership A is: x + y = 164.

An equation for the total cars and trucks for dealership B is: 2x + 0.5y = 229.

The number of cars that dealership A sold is: 98 cars.

The number of trucks that dealership B sold is: 66 trucks.

In order to write a system of linear equations to describe this situation, we would assign variables to the number of cars sold and number of trucks sold, and then translate the word problem into an algebraic equation as follows:

Since the first dealership sold a total of 164 cars and trucks, a linear equation to model this situation is given by;

x + y = 164.

Additionally, the second dealership sold twice as many cars and half as many trucks as the first dealership, with a total of 229 cars and trucks;

2x + 0.5y = 229.

By solving the systems of linear equations simultaneously, we have:

2x + 0.5(164 - x) = 229

1.5x + 82 = 229

x = 98 cars.

For the y-value, we have;

2(98) + 0.5y = 229

0.5y = 229 - 196

y = 33/0.5

y = 66 trucks.

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Apologies, but your questions seem to have errors and incomplete information.

For the first question about Jessica and Melissa sharing 12 pieces of dried pears, it is unclear how much Jessica ate as the fraction is missing. Similarly, information about how many pears Melissa ate is also missing. To answer the question accurately, I need the missing information.

For the second question about the children playing in the park, you mentioned that one-third of the children returned home, but you didn't provide the total number of children initially in the park. Without that information, I cannot determine how many children stayed in the park.

Please provide complete and accurate information for a precise answer.

The equation of a line that passes through points (2,5) and (4,3) is

y = -x+7.

Finding the equation of a line:

First, we need to find out the slope for the given points.

(X1,Y1) = (2,5)

(X2,Y2) = (4,3)

formula for slope(m) = [tex]\frac{Y2 - Y1}{X2 - X1}[/tex]

substitute the points in the above formula

[tex]\frac{3 - 5}{4 - 2}[/tex] = [tex]\frac{-2}{2}[/tex]

[tex]\frac{-2}{2}[/tex] = -1

slope for the given points(m) = -1.

m = -1

The equation of a line is y-y1 = m(x-x1), where x and y are variables.

substituting the values in the above equation then :

y-5 = -1(x-2)

y-5 = -x+2

y+x = 2+5

x+y = 7

y = -x+7

Therefore, the equation of the line passing through the points (2,5) and (4,3) is y = -x+7

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The remaining balance on a CD with a $2,200 initial investment and a 2.95% interest rate after three years would be roughly $2,393.10.

The compound interest formula is as follows:

Balance = Principal * (Rate/100 + Rate * 1)Time

Where

Principal (P) is equal to $2,200.

Rate (R) = 2.95% = 2.95/100 = 0.0295 in decimal form.

Time (T) = 3 years.

The following formula is substituted for the values:

Balance =[tex]$2,200 * (1 + 0.0295)^3.[/tex]

Exponent calculation:

Balance = [tex]$2,200 * (1,0295)^3[/tex]

Calculating the result:

Balance ≈ $2,200 * 1.089135

Balance ≈ $2,393.10

So, the remaining balance on a CD with a $2,200 initial investment and a 2.95% interest rate after three years would be roughly $2,393.10.

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Based on the given data, the relationship appears to be exponential.

To determine if the relationship between the given values is linear, exponential, or neither, we can examine the pattern in the data.

If the relationship is linear, we would expect the values of y to increase or decrease at a constant rate as the values of x increase. In other words, there would be a constant slope between the points.

If the relationship is exponential, we would expect the values of y to change by a constant factor as the values of x increase. In this case, the relationship would exhibit exponential growth or decay.

Looking at the given values of x and y, we can observe the following differences between consecutive x-values: 7, 9, 9, 9. These differences are not constant, indicating that the relationship is not linear.

To determine if the relationship is exponential, we can examine the ratios between consecutive y-values: 5/10 = 0.5, 10/20 = 0.5, 20/40 = 0.5. These ratios are constant, suggesting that the relationship may be exponential with a common ratio of 0.5.

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She sold 27 appliances.

Given,

Lucy earns $400 a month in salary and she receives a commission of $18.

Last Month income = $886

Now,

Equation:

Monthly income = Fixed income + income from commission

$886 = $400 +  income from commission

Income from commission = $486

Commission for one appliance = $18

So,

Total items sold for the commission of  $486,

$486/$18

= 27

Hence total 27 appliances lucy sold.

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The mean of the given set of numbers is 11.

To find the mean (average) of a set of numbers, we sum up all the numbers and divide the sum by the total count of numbers.

Given the set of numbers: 8, 14, 22, 7, 2, 11, 25, 7, 5, 9

To find the mean, we add up all the numbers:

8 + 14 + 22 + 7 + 2 + 11 + 25 + 7 + 5 + 9 = 110

Next, we divide the sum by the total count of numbers, which is 10:

110 / 10 = 11

Therefore, the mean of the given set of numbers is 11.

The mean is a measure of central tendency and represents the average value of the data set.

In this case, it indicates that, on average, the numbers in the set tend to cluster around the value of 11.

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As per the given triangle, the base of the triangle, b, is 360 millimeters.

We may use the formula for the area of a triangle to determine the triangle's base, or b, given its 360 square millimetres in area, 41 mm for each side, and 40 millimetres in height.

The formula for calculating a triangle's area is:

Area = (base * height) / 2

Given that,

Area = 360 square millimetres

Height = 40 mm

Sides = 41 mm

So, 360 = (b * 40) / 2

720 = b * 2

b = 720 / 2

b = 360

Thus, the base of the triangle is 360 mm.

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Answer:

y = -1/2x

Step-by-step explanation:

Currently, y = 2x - 5 is in slope-intercept form, whose general equation is y = mx + b, where

The slopes of perpendicular lines are negative reciprocals of each other.  We can show this with the following formula;

m2 = -1 / m1, where

Finding the slope of the other line:

Since 2 is the slope of y = 2x - 5, we can plug in 2 for m1 in perpendicular slope formula to find m2, the slope of the other line:

m2 = -1 / 2

m2 = -1/2

Thus, the slope of the other line is m = -1/2.

Finding the y-intercept of the other line:

We can find the y-intercept of the other line (i.e., b in the slope-intercept equation) by plugging in (-2, 1) for x and y and -1/2 for m in the slope-intercept form:

1 = -1/2(-2) + b

1 = 1 + b

0 = b

Thus, y = -1/2x is the line that passes through (-2, 1) and is perpendicular to y = 2x - 5.

We can integrate this S = 2π ∫(1.1 to 4.4) (5√(4x + 25))/(2√x) dx over the given interval (1.1 to 4.4) to find the surface area.

We can evaluate the integral using numerical methods or a calculator to find the final answer.

We have,

To find the surface area of the revolution about the x-axis of the function f(x) = 5√x over the interval (1.1 to 4.4), we can use the formula for the surface area of revolution:

S = ∫(a to b) 2πy√(1 + (f'(x))²) dx

In this case,

f(x) = 5√x, so f'(x) = (d/dx)(5√x) = 5/(2√x).

Let's calculate the surface area:

S = ∫(1.1 to 4.4) 2π(5√x)√(1 + (5/(2√x)²) dx

Simplifying the expression inside the integral:

S = ∫(1.1 to 4.4) x 2π(5√x)√(1 + 25/(4x)) dx

Next, we can integrate this expression over the given interval (1.1 to 4.4) to find the surface area.

To find the surface area of revolution about the x-axis of the function

f(x) = 5√x over the interval (1.1 to 4.4), we need to evaluate the integral:

S = ∫(1.1 to 4.4) 2π(5√x)√(1 + 25/(4x)) dx

Let's calculate the integral:

S = 2π ∫(1.1 to 4.4) (5√x)√(1 + 25/(4x)) dx

To simplify the calculation, let's simplify the expression inside the integral first:

S = 2π ∫(1.1 to 4.4) (5√x)√((4x + 25)/(4x)) dx

Next, we can distribute the square root and simplify further:

S = 2π ∫(1.1 to 4.4) (5√(4x + 25))/(2√x) dx

Thus,

We can integrate this expression over the given interval (1.1 to 4.4) to find the surface area.

We can evaluate the integral using numerical methods or a calculator to find the final answer.

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The algebraic expression for the sentence in this problem is given as follows:

n - (x + 8).

The sentence in this problem is given as follows:

"A number n minus the sum of the number and eight".

The sum of an unknown number x and eight is given as follows:

x + 8.

The subtraction of the number n by the expression is given as follows:

n - (x + 8).

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