Solve the system of equations by multiplying first. Enter the solution as an ordered pair.



2x + 8y = 36

5x − 4y = 18


the ordered pair solution


(?,?)

Paul's pig pen has a width of 2 feet and an area of 16 square feet is the true statement.

The area of Rectangle is length times of width.

This is a simple calculus problem but it can be solved without calculus, as follows:

let L=length of the pen and W = width

2(L + W) = 20  

L + W = 10

L = 10 - W

Let A = area of the pen

A = Length × Width = 10W - W²

Sketch this parabola which has a positive maximum which occurs at W = 10/2

This means the area is maximum when w = 2, in other words the maximum area occurs when the rectangle is a square.

Hence,  Paul's pig pen has a width of 2 feet and an area of 16 square feet.

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The equation that is graphed is (c) y = x² + 2    if x < 1 and y = -x + 2  if x ≥ 1

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

The graph (see attachment)

On the graph, we can see that the first curve is a quadratic function,

Also, it has an open circle at x = 1

So, we have:

x² + 2    if x < 1

Next, we have a linear equation with a closed circle at x = 1

So we have:

y = -x + 2 for x ≥ 1

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Answer: The graph of f(x) = 2^x and g(x) = 2^x +2 are the same.

Explanation: The functions f(x) = 2^x and g(x) = 2^x + 2 are identical. The only difference between them is the constant term in the second function. This constant term does not change the shape, direction or the relative position of the graph of the function, it only shifts the graph along the y-axis. Because both functions have the same base, the graphs will have the same shape, direction and relative position, regardless of the constant term.

Answer:

f(g(0)) = 12

Step-by-step explanation:

to find f(g(0)) , evaluate g(0), then substitute the value obtained into f(x)

g(0) = 0 + 4 = 4 , then

f(4) = 3(4) = 12

The x-intercept of (-2, 0) has both functions. The behavior of the function g(x) for x ⇒ -∞ then g(x) → ∞. The y-intercept of the function f(x)  is (0, 6). And both functions have real roots.

A function is an assertion, concept, or principle that establishes an association between two variables. Functions may be found throughout mathematics and are essential for the development of significant links.

The functions are given below.

f(x) = x³ - 2x² - 5x + 6

g(x) = - x³ + 4x

The factor of the function f(x) is given as,

f(x) = 0

x³ - 2x² - 5x + 6 = 0

(x + 2)(x - 1)(x - 3) = 0

x = -2, 1, 3

The factor of the function g(x) is given as,

g(x) = 0

- x³ + 4x = 0

x = 0, 2, -2

The y-intercept of the function f(x) is given as,

f(0) = (0)³ - 2(0)² - 5(0) + 6

f(0) = 6

The y-intercept of the function g(x) is given as,

g(0) = - (0)³ + 4(0)

g(0) = 0

The graph of the function is given below.

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1 gallon of water must be added to 7 gallons of an 8% insecticide solution to reduce the concentration to 7%.

Let x = amount of water required

7 gallons + x gallons = (x + 7) gallons of mixture

For x gallons of water, we have 0 gallons of insecticide

For 7 gallons of 8% insecticide solution, we have 0.08(7)=0.56 gallons of insecticide

According to the question, we get the equation as follows

0.07(x + 7) = 0.56

7(x + 7) = 56

x + 7 = 8

x = 1 gallon

Therefore, the amount of water that needs to be added to 7 gallons of an 8% insecticide solution to reduce the concentration to 7% is 1 gallon.

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A perfect square is a number obtaining by squaring a whole number

A perfect square can be defined as a number that is being expressed as the product of an integer by itself.

It is also seen as the second exponent of an integer.

In determining the perfect of a number, when you multiply an integer which could be a “whole” number, positive, zero or negative) by itself, the product of the multiplication is termed a square number, or a perfect square.

Perfect square is also described as a number made by squaring a whole number.

Some perfect squares in mathematics are;

Number 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 139, 144, etc

Hence, a perfect square is described as a number that is the product of an integer with itself.

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As per the given mortgage, the remaining balance be after the first 5 years is $678125

Here Kathy and Jake are purchasing a house and are financing $465,000. The mortgage is a 20-years 5/1 ARM at 3. 5% with a 2/7 cap structure.

As we know that the formula for the calculation is written as

=> (Mortgage X Rate)/ 12 months

And then we have to subtract the Monthly PI payment from monthly interest on payment.

When we apply the value on it, then we get,

($465,000 x 3.5)/12

= $135625

Then the amount after 5 years is calculated as,

⇒ $135625 x 5

= $678125

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The required monthly payment for 6.4% of interest is $16846.6.

Interest is the cost of borrowing money or the fee you charge to lend it. Most frequently, interest is shown as an annual percentage of the loan amount. The interest rate for the loan is denoted by this proportion. ​

We have,

Purchasing price = $200,000

Down payment = 5% of $200,000

= $10000

Remaining Amount = $190000

Interest of one year = 6.4%

Amount of interest =  6.4% of $190000

Amount of interest = $12160

Total amount = f $190000 + $12160

Total amount = $202160

Monthly payment = $16846.6

Thus, required monthly payment is  $16846.6.

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The total cost of Lacey's lesson a function of the number of hours scheduled is [tex]f(x)=\left \{ {{(10 + 5)x, x\leq 3} \atop {10x, x > 3}} \right.[/tex] and this one refers the function.

In math the term called function is known as a correspondence from one value x of the first set A to another value y of the second set B and it relates inputs to outputs.

Here we have know that Lacey pays $15 for each hour of golf lesson and $10 for equipment rental.

And here we have also know that If the lesson is more than 3 hours long she only pays $5 for equipment rental.

Let us consider that x be the amount time she has been taking lessons.

Then the function can be written as,

[tex]f(x)=\left \{ {{(10 + 5)x, x\leq 3} \atop {10x, x > 3}} \right.[/tex]

While we looking into it we have identified that the given situation will create the function.

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This statement is False.

The formula provided in the statement is used to calculate the variance of the sample mean difference between two independent samples, but it is not necessarily true that the sample variances, and , are independent of one another.

The sample variances are calculated based on the data from their respective samples, and thus their relationship will depend on the specific data and the underlying population distributions.

For example, let's consider two samples of size n = 25, one from a normal population with mean 100 and variance 10, and another one from a normal population with mean 110 and variance 20. The sample variances for each sample are:

S1² = 9.6 and S2² = 20.8

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Mean is 27 and SD is 2 then the normal curve ​with highest range would be at domain 27.

The mean is the average of a set of numbers. It is calculated by adding all the numbers in the set together, and then dividing by the number of values in the set.  It is calculated by adding up all the numbers in the set, dividing by how many values there are in the set, and then multiplying by that number. The average of a group of numbers is called the mean.

Mean is 27 and SD is 2

a) P(23<x<27)= P((23-27)/2<z<(27-27)/2)=P(-2<z<0)=P(z<0)-P(z<-2) or P(z<0)-(1-P(z<2))

From normal distribution table we get 0.5-(1-0.9772)=0.4772

b) P(x>31)=P(z>(31-27)/2)=P(z>2) or 1-P(z<2)=1-0.9772=0.0228

c) P(x<29)= P(z<(29-27)/2)=P(z<1)=0.8413

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The range of the function is (d) [150, 275]

The graph that completes the question is added as an attachment

From the graph, we have the following properties:

The range is simply the amount of values between the y axis

In this case represented by the graph, we have:

Minimum y value = 150

Maximum y values = 275

Sinfe both values are inclusive, the range would be [150, 275]

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The number of terms in the algebraic expression 6y + 14 is 2 terms

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

6y + 14

Consider an expression given as

ax + b

The number of terms in the expression is 2 and the terms are ax and b

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

6y + 14 has 2 terms

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Complete question

Identify the number of terms and then the resend themselves for each algebraic expression

6y + 14

The bar representing her 7-day vacation will have a class width of 16 - 22 and a height of 1.

The frequency distribution of a collection of data reveals how frequently each unique value appears.

The most typical graph for displaying frequency distributions is a histogram. Although it closely resembles a bar chart, there are significant variances.

In the given histogram class width(horizontal width)  represents the number of days and frequency(vertical height) represents the number of vacations taken.

Therefore, The 7-day vacation yet she took this summer will be represented by a bar having a class width 16 - 22 and a height of 1.

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The answer of [tex]\sqrt[4]{125a^{4} }[/tex] is 3.34a. The solution has been obtained by using the concept of exponents.

What is an exponent?

The exponent of a number shows that how many times a particular number has been multiplied by itself. For instance, the number 2^4 means that we are multiplying the number 2 itself by four times. It is enlarged to 2*2*2*2. An exponent is also known as a number's power. One may use a whole number, fraction, negative number, or decimal.

We are given [tex]\sqrt[4]{125a^{4} }[/tex]

This can be also written as (125a^4)^1/4

Splitting the power, we get

⇒[tex]a^{4^{1/4} } 125^{1/4}[/tex]

⇒a 125^0.25

⇒a 3.34

⇒3.34a

Hence, the answer of [tex]\sqrt[4]{125a^{4} }[/tex] is 3.34a.

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Since there are multiple questions so, the question answered above is

[tex]\sqrt[4]{125a^{4} }[/tex]

Answer: 3.6e+18

Step-by-step explanation: (400 x 9 = 3,600) 10 to the power of 15= 1,000,000,000,000,000 or quadrillion. 9x1,000,000,000,000,000=9e+15x400=3.6e+18

After solving the value of y is 541.

The value of set A is 281, 268, 225, 290. So

The mean of set A = (281 + 268 + 225 + 290)/4

The mean of set A = 1064/4

The mean of set A = 266

The value of set B = 392, 512, 413, 422, у

The mean of set B = (392 + 512 + 413 + 422 + у)/5

The mean of set B = (1739 + y)/5

The ratio of Mean of Set A : Mean of Set B = 7 : 12

So, 7x = 266

Divide by 7 on both side, we get

x = 266/7

x = 38

Now, 12x = 12 × 38 = 456

The mean of set B = 456

(1739 + y)/5 = 456

Multiply by 5 on both side, we get

1739 + y = 2280

Subtract 1739 on both side, we get

y = 541

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The complete question is:

Here are two sets of numbers, A and B.

 Set A                            Set B

281  268                     392  512  413

225  290                        422  у

Mean of Set A : Mean of Set B = 7 : 12.

Work out the value of y.

Answer:

w = 11 , x = 25

Step-by-step explanation:

for Δ ABC ≅ Δ TUV, then corresponding sides must be congruent, that is

AB = TU and BC = UV ( using these to create 2 equations )

- 9x + 20w + 89 = 4w + x + 15 ( subtract 4w + x + 15 from both sides )

- 10x + 16w + 74 = 0 → (1)

and

x + w + 41 = - 18w + 11x ( subtract - 18w + 11x from both sides )

- 10x + 19w + 41 = 0 → (2)

multiplying (2) by - 1 and adding to (1) will eliminate x

10x - 19w - 41 = 0 → (3)

add (1) and (3) term by term to eliminate x

0 - 3w + 33 = 0 ( subtract 33 from both sides )

- 3w = - 33 ( divide both sides by - 3 )

w = 11

substitute w = 11 into either of the 2 equations and solve for x

substituting into (1)

- 10x + 16(11) + 74 = 0

- 10x + 176 + 74 = 0

- 10x + 250 = 0 ( subtract 250 from both sides )

- 10x = - 250 ( divide both sides by - 10 )

x = 25

thus the values x = 25 and w = 11 will make the triangles congruent

The length of Mian in inches is 23 and the length of Udon in inches is 11

An equation is a mathematical statement that shows that two mathematical expressions are equal.

Given that, Aria has two pet snakes, Mian and Udon. The combined length of the two snakes is 34 in and Udon is 12 inches shorter than Mian.

Let Mian be represented by M and that of Udon by U,

Therefore, according to question, establishing the equation,

M + U = 34

M - U = 12

Adding both the equations,

2M = 46

M = 23

U = 11

Hence, the length of Mian in inches is 23 and the length of Udon in inches is 11

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If the vehicle has 10 gallons of fuel left, it has driven 600 miles.

The term "equation" refers to mathematical statements with at least two terms that contain variables or integers that are equal.

The amount of gas in the tank should be x gallons, and the distance travelled by car should be d miles.

Due to the car's 16-gallon fuel tank

So, 16 - x would be the remaining fuel in the tank.

It gets 85 miles per gallon of gas when travelling on a highway.

⇒ d = (16-x)85

if the tank still holds 10 gallons of fuel

In the equation above, if x = 10, we get

⇒ d = (16-10)85.

⇒ d = (20)85.

⇒ d = 600.

If the vehicle has 10 gallons of fuel left, it has driven 600 miles.

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The number of hamsters as a function of the cycle number, c, can be represented by the equation:

f(c) = 5 * 3^c

This equation was determined by the fact that the population triples every cycle and the starting number of hamsters is 5. The function notation is f(c) which represents the number of hamsters as a function of the cycle number. The base of the equation is 3, which represents the rate at which the population triples. The exponent is c, which represents the cycle number. The coefficient is 5, which represents the starting number of hamsters.

Parallelograms are quadrilaterals with two sets of parallel sides

Slope Formula: y2-y1/x2-x1

A quadrilateral with two sets of parallel sides is referred to as a parallelogram. In a parallelogram, the opposing sides are of equal length, and the opposing angles are of equal size. Additionally, the interior angles are additional to the transversal on the same side. 360 degrees is the sum of all interior angles.

KL: 12-7/6-2 = 5/4

LM; 13-12/13-6 = 1/7

MN: 8-13/9-13 = 5/4

KN: 7-8/9-2 = 1/7

We can identify this as a parallelogram since it has congruent angles and sides, which are found on a parallelogram.

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The statement in algebraic terms is given by 8z-89/8

An algebraic expression in mathematics is an expression which is made up of variables and constants, along with algebraic operations (addition, subtraction, etc.). Expressions are made up of terms.

Given here: The statement " 11¹/₈less than the product of 8 and the number z"

The translation of the statement in the form of an algebraic expression is given by

8z - 11¹/₈

8z-89/8

Hence, The statement in algebraic terms is given by 8z-89/8

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The slope and the intercept of the function f(x) = -x + 5 are given as follows:

The slope-intercept definition of a linear function is given as follows:

y = mx + b.

For which the parameters are given as follows:

The function for this problem is defined as follows:

f(x) = -x + 5.

Comparing to the standard definition of a linear function, we have that:

The problem asks for the slope and the intercept of the function f(x) = -x + 5.

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(a) The study is observational type.

(b) Stratified sampling can be used for carrying out this study.

(a) In the given problem, We want to hear students' opinions about the new housing structure and ensure that the survey reflects all students' views equally. The critical thing to note here is that you need to collect opinions. Not the reason behind the opinion. So researchers don't care how the data goes up. So this is an observational study.

(b) In the case of sampling strategy we can use stratified sampling. Stratified sampling refers to a type of sampling technique. In stratified sampling, researchers divide the population into separate groups called strata. A probability sample (often a simple random sample) is then drawn from each group. The same number of samples can be used each year.

Therefore, To make sure the survey equally represents opinions from students from all years (a) the study is the observational type and (b) Stratified sampling can be used for carrying out this study.

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The length of all a triangle's sides added together yields the perimeter of a triangle.

The length of all a triangle's sides added together yields the perimeter of a triangle.

The result of the lengths of the sides is the perimeter of any polygon. In the case of a triangle: Perimeter = Sum of the three sides.

perimeter=(length + width)2.

JKL - Delta RST^ prime , RS = 14 ST = 12 , TR = 10 , and LJ = 14 2. ADEF, if triangle DEF sim triangle ABC , AB = 27 BC = 16 CA = 25 , and FD = 15 3

RST perimeter=(length + width)2.

14+ 12+10+14 = 50

triangle ABC

27 + 16 + 25 =68

triangle PQR sim triangle LMN , LM = 16 , MN = 14 , NL = 27 ,

16+14 + 27 = 57

if triangle KTM sim triangle FGH . . FG = 30 GH = 38 , HF = 38

30 + 38 + 38 =106

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The monthly payment during the initial period is  about $1,682.94

The type of loan Jackie obtains is an Adjustable Rate Mortgage, ARM, loan, which is a 4/2 ARM

The introductory interest rate of 5% is fixed for the first 4 years, and will be adjusted every 2 years after the first four years.

Let us assume that m represents the initial monthly payment.

[tex]M=\frac{P.(\frac{r}{12} ).(1+\frac{r}{12} )^n}{(1+\frac{r}{12} )^n - 1}[/tex]

where, M is the monthly payment during the initial period

P is the amount on loan

r is the interest rate = 5% = 0.05

n is period = 30 × 12 = 360

Here, P = $313,500

r = 5%

r = 0.05

n =  30 × 12

n = 360

Substituting these values in the above formula we get,

[tex]M=\frac{313500\times (\frac{0.05}{12} ).(1+\frac{0.05}{12} )^{360}}{(1+\frac{0.05}{12} )^{360} - 1}[/tex]

M  ≈ 1,682.94

Therefore, the monthly payment is about $1,682.94

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The long division problem can be used to prove the formula for factoring the difference of two perfect cubes (a-b)(a²+ab+b²).

The difference of two perfect cubes: a³-b³  = (a-b)(a²+ab+b²)

let the two perfect cubes be a³ and b³. Factoring the  difference between these two perfect cubes we have: a³ - b³

First, we need to factorize (a-b)³:

(a-b)³ = (a-b) (a-b)²

(a-b)³ = (a-b)(a²-2ab+b²)

(a-b)³ = a³-2a²b+ab²-a²b+2ab²-b³

(a-b)³ = a³-b³-2a²b-a²b+ab²+2ab²

(a-b)³ = a³-b³ - 3a²b+3ab²

(a-b)³  = (a³-b³) -3ab(a-b)

Then we will make a³-b³ the subject of the formula from the result equation;

a³-b³ =  (a-b)³+ 3ab(a-b)

a³-b³  = a-b{(a-b)²+3ab}

a³-b³  = a-b{a²+b²-2ab+3ab}

a³-b³  = (a-b)(a²+b²+ab)

a³-b³  = (a-b)(a²+ab+b²)

The long division problem that can be used is (a-b)(a²+ab+b²).

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