f(x) = x − 3 and g(x) = 2x + 8 Which ordered pair is a solution for the system?

Final answer:

The square root of -9 is not -3 but 3i, indicating the need for complex numbers to describe the square root of negative numbers, with 'i' denoting the imaginary unit.

Explanation:

The value of the square root of -9 is not -3 because the square root operation defined for real numbers does not apply to negative numbers in the same way. Instead, when dealing with the square root of a negative number, we enter the realm of complex numbers.

A complex number is a combination of a real part and an imaginary part. The square root of -1 is defined as i, the imaginary unit. Therefore, the square root of -9 is correctly expressed as 3i, not -3. This is a critical distinction because the result provides a basis for complex arithmetic, expanding the scope of mathematical operations into the complex plane.

Complex numbers and imaginary numbers are fundamental in various fields of science and engineering, particularly in situations involving quadratic equations with no real roots or in the analysis of electrical circuits. Recognizing the utility of the imaginary unit, i, allows for solving equations that otherwise would not have solutions within the set of real numbers.

[tex]\dfrac{sin\theta + cos\theta}{sin\theta-cos\theta}+\dfrac{sin\theta-cos\theta}{sin\theta+cos\theta}=\dfrac{2sec^2\theta}{tan^2\theta-1}[/tex]

From Left side:

[tex]\dfrac{sin\theta + cos\theta}{sin\theta-cos\theta}\bigg(\dfrac{sin\theta+cos\theta}{sin\theta+cos\theta}\bigg)+\dfrac{sin\theta-cos\theta}{sin\theta+cos\theta}\bigg(\dfrac{sin\theta-cos\theta}{sin\theta-cos\theta}\bigg)[/tex]

[tex]\dfrac{sin^2\theta+2cos\thetasin\theta+cos^2\theta}{sin^2\theta-cos^2\theta}+\dfrac{sin^2\theta-2cos\thetasin\theta+cos^2\theta}{sin^2\theta-cos^2\theta}[/tex]

NOTE: sin²θ + cos²θ = 1

[tex]\dfrac{1 + 2cos\theta sin\theta}{sin^2\theta-cos^2\theta}+\dfrac{1-2cos\theta sin\theta}{sin^2\theta-cos^2\theta}[/tex]

[tex]\dfrac{1 + 2cos\theta sin\theta+1-2cos\theta sin\theta}{sin^2\theta-cos^2\theta}[/tex]

[tex]\dfrac{2}{sin^2\theta-cos^2\theta}[/tex]

[tex]\dfrac{2}{\bigg(sin^2\theta-cos^2\theta\bigg)\bigg(\dfrac{cos^2\theta}{cos^2\theta}\bigg)}[/tex]

[tex]\dfrac{2sec^2\theta}{\dfrac{sin^2\theta}{cos^2\theta}-\dfrac{cos^2\theta}{cos^2\theta}}[/tex]

[tex]\dfrac{2sec^2\theta}{tan^2\theta-1}[/tex]

Left side = Right side so proof is complete

Answer:

\dfrac{sin\theta + cos\theta}{sin\theta-cos\theta}+\dfrac{sin\theta-cos\theta}{sin\theta+cos\theta}=\dfrac{2sec^2\theta}{tan^2\theta-1}

From Left side:

\dfrac{sin\theta + cos\theta}{sin\theta-cos\theta}\bigg(\dfrac{sin\theta+cos\theta}{sin\theta+cos\theta}\bigg)+\dfrac{sin\theta-cos\theta}{sin\theta+cos\theta}\bigg(\dfrac{sin\theta-cos\theta}{sin\theta-cos\theta}\bigg)

\dfrac{sin^2\theta+2cos\thetasin\theta+cos^2\theta}{sin^2\theta-cos^2\theta}+\dfrac{sin^2\theta-2cos\thetasin\theta+cos^2\theta}{sin^2\theta-cos^2\theta}

NOTE: sin²θ + cos²θ = 1

\dfrac{1 + 2cos\theta sin\theta}{sin^2\theta-cos^2\theta}+\dfrac{1-2cos\theta sin\theta}{sin^2\theta-cos^2\theta}

\dfrac{1 + 2cos\theta sin\theta+1-2cos\theta sin\theta}{sin^2\theta-cos^2\theta}

\dfrac{2}{sin^2\theta-cos^2\theta}

\dfrac{2}{\bigg(sin^2\theta-cos^2\theta\bigg)\bigg(\dfrac{cos^2\theta}{cos^2\theta}\bigg)}

\dfrac{2sec^2\theta}{\dfrac{sin^2\theta}{cos^2\theta}-\dfrac{cos^2\theta}{cos^2\theta}}

\dfrac{2sec^2\theta}{tan^2\theta-1}

Left side = Right side so proof is complete

Step-by-step explanation:

Final answer:

The solution to the system of equations is (-6, 8), found by substituting x = -6 into the first equation and simplifying to find the value of y.

Explanation:

The solution to the system of equations y= -1/3x+6 and x = -6 can be found by substituting the value of x from the second equation into the first equation. Here's how it's done:

Substitute x into the first equation: y = -1/3(-6) + 6.

Simplify the equation: y = 2 + 6.

Add the numbers to find the value of y: y = 8.

Therefore, the solution to the system of equations is the point (-6, 8).

Answer:

8m^4  -8m^3  -7m^2 -12 m

Step-by-step explanation:

(4m^4 - 8m^3 - 7m) + (4m^4 - 7m^2 - 5m)

I like to line them up vertically.  I put 0 in the spots to keep them lined up

4m^4 - 8m^3  +0m^2  -7m

+4m^4 +0m^3 - 7m^2 - 5m

--------------------------------------------

8m^4  -8m^3  -7m^2 -12 m

[tex](4m^4-8m^3-7m)+(4m^4-7m^2-5m)\\\\=4m^4-8m^3-7m+4m^4-7m^2-5m\qquad\text{combine like terms}\\\\=(4m^4+4m^4)-8m^3-7m^2+(-7m-5m)\\\\=\boxed{8m^4-8m^3-7m^2-12m}[/tex]

Answer:

see below

Step-by-step explanation:

In a right triangle, sin = opposite /hypotenuse

 but m is adjacent to the angle, so we need to use cosine

cos = adjacent /hypotenuse

cos 73 = m/9

Multiply each side by 9

9 cos 73 = m

2.631345343 =m

Answer:

In a 28 day month 20 dollars and 30 cents

Step-by-step explanation:

1+2+3+4+5+6+7......

Answer*5=Cents

If cents equal to or greater than 100 than = dollar

$30 money you would have if for a month you were given a nickel on day one, two nickels on day two, three nickels on day three, and so on.

The arithmetic series is a series in which the numbers are in the order of some common difference. The difference between any two consecutive numbers is constant.

[tex]a_n = a_1 + (n -1)d[/tex]

you would be using a be using the [tex]a_n = a_1 + (n -1)d[/tex] meaning that it's the previous entry plus the different process.

[tex]a_n = a_1 + (n -1)d[/tex]

[tex]a_{30} = .05 + (30-1).05[/tex]

[tex]a_{30}[/tex] = $29.10

Thus, $30 money you would have if for a month you were given a nickel on day one, two nickels on day two, three nickels on day three, and so on.

Learn more about arithmetic series:

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Answer:

From the table: you have to have the same increment for the x values and the same increment for the y values. Doesn't have to be the same increment for both x and y values.

From an equation: the exponent of the x and y should be 1. The equation should be a line equation.

From a graph: the graph should be a continuous straight line.

Step-by-step explanation:

Answer:

$45.61

Step-by-step explanation:

1) On the first day of the month, John counted his money and found he had $28.35

2) He has received an additional $17.26

so

to find the total money he has now sum the two quantities

$28.35 + $17.26 = $45.61

therefore

the answer is

$45.61

f(x) + n - move the graph n units up

f(x) - n - move the graph n units down

n · f(x) - stretch by n in the y-direction

1/n · f(x) - compress by n in the y-direction

-------------------------------------------------------------------------

First:

[tex]A)\ y=x^2\\\\B)\ y=x^2-4\\\\C)\ y=x^2-2\\\\D)\ y=x^2+2\\\\E)\ y=x^2+4[/tex]

Second:

[tex]A)\ y=x^2\\\\K)\ y=4x^2\\\\L)\ y=2x^2\\\\M)\ y=\dfrac{1}{2}x^2\\\\N)\ y=\dfrac{1}{4}x^2[/tex]

Final answer:

The weekly take-home pay from a $45,000 annual salary with a 22% total tax burden is approximately $675.00 after taxes.

Explanation:

To calculate the take-home pay from a paycheck when you know the annual salary and the total tax rate, you must first determine the amount of money deducted for taxes and then subtract that from the gross (before-tax) pay. If a job pays $45,000 per year and the total tax burden is 22%, we can find the annual take-home salary and then divide by the number of pay periods per year to get the weekly take-home pay.

First, calculate the total annual taxes: $45,000 * 0.22 = $9,900.

Next, subtract the annual taxes from the annual salary to get the annual take-home salary: $45,000 - $9,900 = $35,100.

Since there are 52 weeks in a year, divide the annual take-home salary by 52 to find the weekly take-home pay: $35,100 / 52 \\approx $675.00.

Therefore, the weekly take-home pay would be approximately $675.00.

Answer:

a^2+4a-2

Step-by-step explanation:

f(x)= x^2+2x-5

f(a+1) = (a+1)^2 + 2(a+1) -5

        = a^2 + 2a + 1 + 2a +2 -5

       =a^2+4a -2

Answer:

4+y

Step-by-step explanation:

4 · y

4 + y

4y

(4)(y)

All of these are multiplication except 4+y

Ursula could have drawn a pentagon because all angles of a pentagon are obtuse angles.

An angle greater than 90° is called an obtuse angle.

Trapezoid: A quadrilateral with at least one pair of parallel sides is called

a trapezoid.

In a trapezoid, all the angles are not greater than 90° i.e. all the angles are not obtuse angles.

Similarly, In triangles and parallelograms, all the angles are not obtuse angles.

Pentagon: In a pentagon, all angles are greater than 90° i.e. obtuse angles.

Hence, Ursula could have drawn a pentagon because all angles of a pentagon are obtuse angles.

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Answer:

The answer is option A.

True

Hope this helps you

Answer:

Mean of resulting data = 7 times mean of original data=7a

Step-by-step explanation:

Given that all values in a data set are multiplied by 7. to find the mean of the new set.

Let {x1,x2....xn} be the original data set.

The mean of this data set a= [tex]\frac{x1+x2+...+xn}{n}[/tex]

Now multiply all values by 7

New data set

= {7x1, 7x2...7xn}

Sum of these = 7(x1+x2+..+xn) = 7 times sum of original data set

No of element = n for both the sets

Hence new mean = 7 * mean of original set.=7a

in other words, if the mean of the values in a data set is a. If each of the values in the data set were multiplied by 7, new mean would be 7a.

Answer:

It’s 7a

Step-by-step explanation:

Calculate total area:

12 x 10 = 120 square feet

Convert square feet to square meters:

1 square foot = 0.0929 square meters

120 x 0.0929 = 11.1484 square meters

Total cost = 11.1484 x 25 = $278.71

Answer: It would be $4.38

Step-by-step explanation: If you divide the $5 by .8 it will give you what 1/8 of it would be then you multiply it by 7 to get the 7/8 of it.

I think it's Ten thousandths, zero eighty-eight.

I'm not too sure on this, but hopefully i helped you.

Answer:

The midpoint is (4,6)

Step-by-step explanation:

When we have 2 points, we can find the midpoint by using the formula

midpoint = (x1+x2)/2, (y1+y2)/2

              = (6+2)/2, (4+8)/2

                   = (8/2),  12/2

                = 4,6  

ANSWER:

To prove that opposite sides of a parallelogram are congruent, we have to prove that the quadrilateral is a parallelogram.

The ways to prove that a quadrilateral is a parallelogram include:

- a quadrilateral with parallel opposite sides

- a quadrilateral with congruent opposite angles

- a quadrilateral with diagonals that bisect each other

Answer:

alternate interior angles of congruent

opposite sides are parallel

Step-by-step explanation:

ap3x

Answer:

12.25 or 49/4

Step-by-step explanation:

12.25 is the decimal version because 1/4 is equal to 0.25 as a decimal and then add the 12 on.

13/4 is the improper fraction. To get an improper fraction, you multiply the main number (12) by the denominator (4) this gives you 48. Then, you have to add your numerator (1), leaving you with 49 over 4.

The value of the mixed fraction number 12 ¹/₄ will be 12.25.

When the relevant components and basic processes of a numerical method are given values, the expression's result is the result of the computation it depicts.

The definition of simplicity is making something simpler to achieve or grasp while also making it a little less difficult.

The mixed fraction number is given below.

⇒ 12 ¹/₄

Convert the mixed fraction number into a decimal number. Then we have

⇒ 12 ¹/₄

⇒ 12 + ¹/₄

⇒ 12 + 0.25

⇒ 12.25

The value of the mixed fraction number 12 ¹/₄ will be 12.25.

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Answer:

5

Step-by-step explanation:

700-35w=450+15w

Subtract 15 from -35

700-50w=450

Subtract 700 from 450

-50w=-250

divide -250 by -50w

answer is 5

[tex]700-35w=450+15w\qquad\text{subtract 700 from both sides}\\\\-35w=-250+15w\qquad\text{subtract 15w from both sides}\\\\-50w=-250\qquad\text{divide both sides by (-50)}\\\\\boxed{w=5}[/tex]

Answer:

14th answer is [tex]x=\sqrt{288} \\ x= -\sqrt{288}[/tex]

15th answer is [tex]xy^{2} \sqrt{5}[/tex]

Step-by-step explanation:

14)

triangle is a right triangle, the legs will have the same measure

let the legs be x

x^2 + x^2 = 24^2

2x^2 = 576

x^2 = 288

[tex]x=\sqrt{288} \\ x= -\sqrt{288}[/tex]

15)

[tex]area of the circle[/tex][tex]=\pi r^{2}[/tex]

[tex]5x^{2} y^{4} \pi =\pi r^{2} \\ r^{2} =5x^{2} y^{4} \\ r\sqrt{5x^{2} y^{4} } \\ r=\sqrt{5} xy^{2}[/tex]

Answer:

y = abs (x-5) -4

Step-by-step explanation:

We are shifting the absolute value function

f(x) = |x| to

g(x) =  | x +h| + k

where  h is a shift left if h >  0

                               right if h < 0

 and  k is a shift up if k> 0

                           down if k< 0

Since the graph is shifted down and to the right

k is <0 and h < 0

where k = -4 and h=-5

Answer: 71 numbers apart

Step-by-step explanation: 27 and -54 are 71 numbers apart. In order for 27 to each -54 it must first reach 0. You basically add them while ignoring any negatives. Hope this helped :)

Answer: The end behavior of the given function is,

[tex]f(x)\rightarrow 0[/tex] as [tex]x\rightarrow -\infty[/tex]

And, [tex]f(x)\rightarrow \infty[/tex] as [tex]x\rightarrow \infty[/tex]

Step-by-step explanation:

Since, by the property of the graph of the exponential function,

If an exponential function is, [tex]f(x) = ab^x[/tex]

If b >1 ( increasing function ) then its end behavior is,

[tex]f(x)\rightarrow 0[/tex] as [tex]x\rightarrow -\infty[/tex]

And, [tex]f(x)\rightarrow \infty[/tex] as [tex]x\rightarrow \infty[/tex]

While b < 1 ( decreasing function, then its end behavior is,

[tex]f(x)\rightarrow \infty[/tex] as [tex]x\rightarrow -\infty[/tex]

And, [tex]f(x)\rightarrow 0[/tex] as [tex]x\rightarrow \infty[/tex]

Here, given function is,  [tex]f(x)=5^{x-1}[/tex]

Since 5 > 1

Therefore, the end behavior of the given function is,

[tex]f(x)\rightarrow 0[/tex] as [tex]x\rightarrow -\infty[/tex]

And, [tex]f(x)\rightarrow \infty[/tex] as [tex]x\rightarrow \infty[/tex]

Answer:

The common ratio is 3

Step-by-step explanation:

To determine the common ratio, we take the second term and divide by the first term

9/3 =3

To verify this we take the third term and divide by the second term

27/9 = 3

Answer: Rational numbers are any numbers that can be expressed by a fraction with integers in both the numerator and the denominator. The amount of time and paper it takes to put them into an increasing line depends on how many numbers there are and how big of a range they have.

1.Rewrite improper fractions

An improper fraction is a fraction with a bigger numerator than denominator. These numbers can be difficult to compare. For example, numbers such as 26/7, 31/9, 12/5, 8/3 and 22/13 do not have an obvious highest or lowest term. Rewrite improper fractions as mixed numbers or as numbers with decimals. To turn an improper fraction into a decimal, divide the numerator by the denominator. If the denominator is large or a prime number greater than five, use a calculator. For example, divide the numbers 26/7, 31/9, 12/5, 8/3 and 22/13 to create 3.71, 3.44, 2.4, 2.67 and 1.69.

2.Find the highest and lowest terms

Figure out where your number line starts and ends. Put the lowest term on the far left and the highest term on the far right. Remember, negative numbers are lower than positive numbers, and the greater a negative number's distance from zero, the lower it is.

3.Mark benchmark terms

Draw a line extending through your lowest and highest points. Figure out some major points between these values and mark them on your line. For example, if your lowest number is negative 25 and your highest number is 50, find where zero and 25 would be. Mark these as benchmarks or guides.

4.Fill in the remaining numbers

Using your guides, fill in the rest of the numbers. Start with the lowest values on the left and work your way up.  

Rational and irrational numbers can be ordered from least to greatest by converting them into decimal forms. Once in decimal form, they can be compared and ordered just like regular real numbers.

To order rational and irrational numbers from least to greatest, first you need to understand what these numbers are. Rational numbers are numbers that can be expressed as a ratio of two integers, while irrational numbers cannot be expressed as such ratio and their decimal expansions are non-repeating and non-terminating.

If you are given both rational and irrational numbers, you can start by converting all of the numbers to decimal form if they are not already. Then, you can simply order them from least to greatest, just like how you order regular real numbers.

For example, if you are given the numbers 3/2, √2, and 1. You would convert 3/2 to 1.5 and approximate √2 as 1.414. Then you would order them as 1 (1.0), √2 (1.414), and 3/2 (1.5).

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Answer:

13 more right handed students  PLEASE GIVE BRAINLIEST

Step-by-step explanation:

left handed

3 boys + 4 girls = 7 left handed students

right handed

9 boys + 11 girls = 20 right handed students

20 - 7 = 13 more right handed students

Answer: 13 more right handed girls and boys.

   Hope this helped and also please give brainlist really need it need 2 more if you do me as one only need one more and I need them before the end of toaday.