Given the system: 2x + 3y = 12 2x - y = 4 What is the value of x + y?

Answer:

The room has a width of three metres, and a length of seven metres.

Step-by-step explanation:

This is solvable using a system of equations.  First, we're told that its length is four metres more than its width:

l = w + 4

We're also told that it has a perimeter of 20, and know that the perimeter of a rectangle is equal to two times its length plus two times its length.  We can plug the given perimeter into that relationship, giving us:

20 = 2l + 2w

With that, we can simply substitute our first equation into it, replacing l, and then solve for w:

20 = 2l + 2w

20 = 2(w + 4) + 2w

20 = 2w + 8 + 2w

12 = 4w

w = 3

So the width is three metres.  We also know that the length is four metres more than that, telling us that it's seven metres.

We can check that my calculating the perimeter with those numbers and making sure it still matches 20

2 * 3 + 2 * 7

= 6 + 14

= 20

so the answer is good.

Based on the explanation below, the conclusion that Elan is tired because Elan stayed up late is invalid.

In the question, it is stated that if Elan stays up late, he will be tired the next day.

However, staying up late is not the only thing that could make Elan get tired.

This implies that there are so many other things than can make Elan get tired such as working for long hours.

It is possible that Elan was tired because Elans worked long hours.

By implication, the conclusion is invalid.

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Euler's formula can be used to derive the trigonometric identities.

Euler's formula states that for any real number x, the complex exponential function can be represented as follows:

\(\[e^{ix} = \cos(x) + i\sin(x)\]\)

where \(e\) is the base of the natural logarithm, \(i\) is the imaginary unit, \(\cos(x)\) represents the cosine function, and \(\sin(x)\) represents the sine function. This formula provides a powerful connection between exponential functions and trigonometric functions.

To derive trigonometric identities using Euler's formula, we can start with the exponential form of Euler's formula and manipulate it algebraically. Let's consider the complex exponential function for two angles, \(x\) and \(y\):

\(\[e^{ix} \cdot e^{iy}\]\)

Using the properties of exponents, we can simplify this expression:

\(\[e^{ix} \cdot e^{iy} = e^{i(x + y)}\]\)

Now, applying Euler's formula to both sides, we get:

\(\[\cos(x) + i\sin(x) \cdot \cos(y) + i\sin(y) = \cos(x + y) + i\sin(x + y)\]\)

By comparing the real and imaginary parts of this equation, we obtain the sum and product formulas for cosine and sine:

\(\[\cos(x) \cdot \cos(y) - \sin(x) \cdot \sin(y) = \cos(x + y)\]\[\sin(x) \cdot \cos(y) + \cos(x) \cdot \sin(y) = \sin(x + y)\]\)

These are known as the angle addition formulas, and they form the basis for deriving other trigonometric identities. By manipulating and applying these formulas, we can establish relationships such as double-angle formulas, half-angle formulas, and more.

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The maximum price Matthew can pay for the car, considering his repayment capability and the loan terms, is approximately $126,318.29.

To determine the maximum price Matthew can pay for the car, we need to consider his repayment capability and the terms of the loan.

Matthew can make annual repayments of $35,000. Since the loan term is six years, the total amount he can repay over the loan period is $35,000 multiplied by six, which equals $210,000.

To calculate the maximum price of the car, we need to account for the interest rate of 9.25%. The interest rate represents the cost of borrowing and is applied to the loan amount.

Let's assume the loan amount is denoted by P.

The formula to calculate the future value of a loan with interest is:

FV = P(1 + r)^n

Where:

FV = Future value (total amount repaid)

P = Principal amount (maximum price of the car)

r = Interest rate per period (9.25% or 0.0925)

n = Number of periods (six years)

Since Matthew can repay a total of $210,000 over the loan period, we can set up the equation:

$210,000 = P(1 + 0.0925)^6

Now we can solve for P:

P = $210,000 / (1 + 0.0925)^6

Evaluating this expression, we find:

P ≈ $126,318.29

Therefore, the maximum price Matthew can pay for the car, considering his repayment capability and the loan terms, is approximately $126,318.29.

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The total number of oranges in the basket is 95.

A ratio or value that may be stated as a fraction of 100 is called a percentage. And it is represented by the symbol '%'.

Given:

Of a basket of oranges,

20% of them are defective and 76 are in good condition.

Let x be the total number of oranges.

That means 80% = 76 oranges are in good condition.

20% in decimal form is 0.2.

Applying the percentage formula,

(1 - 0.2)x = 76

0.8x = 76

x = 95

And 20% of 95 = 19

Therefore, the total = 95.

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Answer:

um I think it's f I'm not sure tho

The constant of proportionality (k) is equal to 24. Also, an equation which describes a proportional relationship between time in hours (h) and time in days (d) is h = 24d.

A proportion can be defined as an equation which is typically used to represent (indicate) the equality of two (2) ratios. This ultimately implies that, proportions can be used to establish that two (2) ratios are equivalent and solve for all unknown quantities.

Mathematically, a direct proportion can be represented by the following equation:

y = kx

Where:

In terms of time in hours, a day equals to 24 hours. Therefore, the constant of proportionality (k) is equal to 24.

Based on direct proportion, an equation which describes a proportional relationship between time in hours (h) and time in days (d) is given by:

h = 24d.

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Complete Question:

There is a proportional relationship between time in hours and time in days.

A. What is the constant of proportionality?

B. what equation describes this relationship?

Therefore, the sample size needed is approximately 118. The correct option is d) 117 (since it is the closest whole number to the calculated value).

To determine the sample size needed to provide a margin of error of 2 or less with a 95% probability when the population standard deviation equals 11, we can use the formula:

\(n = (Z * σ / E)^2\)

Where:

n = sample size

Z = Z-score corresponding to the desired confidence level (in this case, Z = 1.96 for a 95% confidence level)

σ = population standard deviation

E = margin of error

Plugging in the given values:

\(n = (1.96 * 11 / 2)^2\)

n ≈ 117.57

Since the sample size must be a whole number, we round up to the nearest whole number:

n ≈ 118

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Answer:

1. Slope is 2

2. slope is 1/4

3. Slope is undefined

4. Slope is -1/2

5. slope is 0

6. Slope is 3/2

Step-by-step explanation:

for example:  the y intercept is -3. The next point is (1, -1) you rose 2 and ran (went over) 1. 2/1 is 2

you do the same for all of them.

Undefined is when it goes continuously downward.

the slope is 0 when it is perfectly horizontal.

I hope this helps

Answer:

B. 7b3 + 4ab + 4a2

Step-by-step explanation:

35b5/5b2 + 20ab3/5b2 + 20a2b2/5b2  

7b3 + 4ab + 4a2

Answer:

B = 7b³ + 4ab + 4a²

Step-by-step explanation:

35b^5 + 20ab³ + 20a²b² by 5b² ÷ 5b²

= 35b^5 + 20ab³ + 20a²b² by 5b²

_________________________

5b²

= 7b³ + 4ab + 4a²

Equation #1:

We move all terms to the left:

-15+4r-(-9)=0

We add all the numbers together, and all the variables

4r-6=0

We move all terms containing r to the left, all other terms to the right

4r=6

r=6/4

r=1+1/2

Equation #2:

We move all terms to the left:

5-y/3-(-1)=0

We add all the numbers together, and all the variables

-y/3+6=0

We multiply all the terms by the denominator

-y+6*3=0

We add all the numbers together, and all the variables

-1y+18=0

We move all terms containing y to the left, all other terms to the right

-y=-18

y=-18/-1

y=+18

Equation #3:

Simplifying

42 = 5x + -8

Reorder the terms:

42 = -8 + 5x

Solving

42 = -8 + 5x

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-5x' to each side of the equation.

42 + -5x = -8 + 5x + -5x

Combine like terms: 5x + -5x = 0

42 + -5x = -8 + 0

42 + -5x = -8

Add '-42' to each side of the equation.

42 + -42 + -5x = -8 + -42

Combine like terms: 42 + -42 = 0

0 + -5x = -8 + -42

-5x = -8 + -42

Combine like terms: -8 + -42 = -50

-5x = -50

Divide each side by '-5'.

x = 10

Simplifying

x = 10

Step-by-step explanation:

hope it's beneficial.

Draw my attention for further explanations.✨✨

The given manufacturing process has a CpK value of 0.4, which indicates that it is not capable of meeting the requirement of producing protein bars with a weight of 17.5 ounces.

Explanation:

The CpK (Process Capability Index) value is a measure of how well a process is capable of meeting the specifications or requirements. It takes into account both the process mean and the variability of the process. A CpK value of 1 indicates that the process is capable of meeting the requirement, while a CpK value less than 1 indicates that the process is not capable of meeting the requirement.

In this case, the process mean is 18 ounces and the standard deviation is 2.5 ounces. To find the CpK value, we need to calculate the upper and lower specification limits. Since the requirement is to produce protein bars with a weight of 17.5 ounces, and the requirement is at -2.5 from the process mean, the lower specification limit would be 18 - 2.5 = 15.5 ounces. The upper specification limit would be 18 + 2.5 = 20.5 ounces.

The CpK value can be calculated using the formula:

CpK = min((USL - mean)/3σ, (mean - LSL)/3σ)

where USL is the upper specification limit, LSL is the lower specification limit, σ is the standard deviation, and mean is the process mean.

Substituting the values, we get:

CpK = min((20.5 - 18)/(3 × 2.5), (18 - 15.5)/(3 × 2.5))

= min(0.4, 0.4)

= 0.4

Since the CpK value is less than 1, the process is not capable of meeting the requirement of producing protein bars with a weight of 17.5 ounces. Therefore, the company needs to improve its manufacturing process to meet the requirements.

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Answer:

1.A

-$92

-$212

1.B

To solve this you set up the equation:  x(8) +12 =y. In this equation x=, the number of paintbrushes and y=, the total cost of the purchase. If you plug in the number of paintbrushes that are being bought you should be able to get your Y(total cost).

Hope this helped!

The value of x is either 3 or -5 based on the distance formula.

What is a co-ordinate system?

In pure mathematics, a coordinate system could be a system that uses one or additional numbers, or coordinates, to uniquely confirm the position of the points or different geometric components on a manifold like euclidean space.

Main body:

according to question

Given points A(-1,4) and B(x,7)

Also AB = 5 cm

Formula of distance = \(\sqrt{(y1-y2)^{2}+(x1 -x2)^{2} }\)

here by using points ,

5 = \(\sqrt{(x+1)^{2} +(7-4)^{2} }\)

taking square on both side ,'

25 = \((x+1)^{2} +3^{2}\)

25-9 = (x+1)²

16 = (x+1)²

taking square root on both sides,

x+1= ±4

x = 4-1 = 3                                  or              x = -4-1 = -5

Hence value of x is either 3 or -5.

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Answer: it is D

Step-by-step explanation:

Using the normal distribution, it is found that there is a 0.0582 = 5.82% probability that their mean repair time is less than 8.9 hours.

The z-score of a measure X of a normally distributed variable with mean \(\mu\) and standard deviation \(\sigma\) is given by:

\(Z = \frac{X - \mu}{\sigma}\)

In this problem, the parameters are given as follows:

\(\mu = 8.4, \sigma = 1.8, n = 32, s = \frac{1.8}{\sqrt{32}} = 0.3182\)

The probability that their mean repair time is less than 8.9 hours is the p-value of Z when X = 8.9, hence:

\(Z = \frac{X - \mu}{\sigma}\)

By the Central Limit Theorem

\(Z = \frac{X - \mu}{s}\)

\(Z = \frac{8.9 - 8.4}{0.3182}\)

Z = 1.57

Z = 1.57 has a p-value of 0.9418.

1 - 0.9418 = 0.0582.

0.0582 = 5.82% probability that their mean repair time is less than 8.9 hours.

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Answer:

The slope is -17 and the y-intercept is 2.

Step-by-step explanation:

Answer:

y

=

17

x

2

−

37

2

The slope-intercept form is

y

=

m

x

+

b

, where

m

is the slope and

b

is the y-intercept.

y

=

m

x

+

b

Find the values of

m

and

b

using the form

y

=

m

x

+

b

.

m

=

17

2

b

=

−

37

2

The slope of the line i

Step-by-step explanation:

Let's write and equation for both cases.

In plan 1, it is 500 plus 3% comission. 3% is equivalent to multiply the amount it sale in the week, "x", by 0.03.

So, the equation for plan 1 is:

For plan 2, there is no fixed value, only comission of 5%, which is equivalent of multiplying the weekly sales, "x", by 0.05, so the equation is:

For both plans to result in the same salary, we have:

So, for both plans to result in the same salary, she have to sale 25,000 in a week.

If two exact objects are thrown from identical catapults, it is expected the catapult with the longer arm length throws the ball farther.

Some of the factors that influence the distance in this case are:

In general, an increased arm length increases the distance. This means that the catapult with a longer arm length will throw the ball farther.

Note: This question is incomplete because there is not enough information about the catapults; due to this, I answered it based on general knowledge.

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Answer:

Natalie's catapult will launch the ball further than Aiden's catapult will

Step-by-step explanation:

Simplified expression for the amount is 5(36+23m) .

An expression is any mathematical statement which consists of numbers, variables and an arithmetic operation between them. In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.

here, we have,

Step 1: Form an equation for the amount of money Troy has received after m months.

If the total months that Troy's grandfather has given him money is m, then the number of months that Troy's mother has given him money is m - 3.

⇒ Total amount in account = 300 + 75m + 40(m - 3)

                                             = 300 + 75m + 40m-120

                                             =180 + 115m

                                            =5(36+23m)

Hence, Simplified expression for the amount is 5(36+23m) .

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Answer:

50.265

Step-by-step explanation:

Area is radius square times pi. Radius is half of diameter .Diameter is 8 inches. Radius is 4 inches. 4 squared is 16. 16 times pi is 50.265

The average time it takes to rent a video tape is 0.141 minutes.

Given data:Customers arrive at a video rental desk at the rate of 12 per minute(Poisson).
Each server can handle 8.15 customers per minute(Poisson). If there are 3 servers, we need to determine the average time it takes to rent a video tape.
Let us assume λ = 12 and μ = 3 × 8.15 = 24.45
Average time it takes to rent a video tape = 1 / (μ - λ/n)
Where, n = number of servers⇒ Average time it takes to rent a video tape = 1 / (24.45 - 12/3)⇒ Average time it takes to rent a video tape = 1 / 8.45⇒ Average time it takes to rent a video tape = 0.1185 minutes = 0.141 rounded to three decimal places.

Thus, the average time it takes to rent a video tape is 0.141 minutes.

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Answer

\( \boxed{\mathsf{3x(x + 3)}}\)

Step by step explanation

\( \mathsf{3 {x}^{2} + 9x}\)

Take 3x as common

⇒\( \mathsf{3x(x + 3)}\)

Hope I helped!

Best regards!

Answer:

Step-by-step explanation:

Factor 3x out of 3x^2=3x(x)-9x=Factor 3x out of 9x=3x(x)+3x(-3)=factor 3x out of 3x(x)+3x(-3). =3x(x-3).

Answer:

292.5

Step-by-step explanation:

65 times 4.5 = 292.5