Solve the system of equations

ax+y=2a
x+ay=1+a^2

Answer:

(4x - 1)^2.

Step-by-step explanation:

16x^2 - 8x + 1

Multiplying the first and last coefficients:

16 * 1 = 16

We need 2 numbers whose product is 16 and whose sum is -8.

These are - 4 and -4.

So we now write:

16x^2 - 4x - 4x + 1

Now factor by grouping :

= 4x(4x - 1) - 1(4x - 1)

= (4x - 1)(4x - 1)

= (4x - 1)^2

This method of factoring quadratics is called the 'ac' method - you split the middle term ( in this case the -8x) into 2 parts.

Answer:

[tex](4x-1)^2[/tex]

Step-by-step explanation:

Given quadratic expression:

[tex]16x^2-8x+1[/tex]

To factor a quadratic in the form [tex]ax^2+bx+c[/tex], find two numbers that multiply to ac and sum to b:

Therefore, two numbers are: -4 and -4

Rewrite the middle term as the sum of these two numbers:

[tex]\implies 16x^2-4x-4x+1[/tex]

Factorize the first two terms and the last two terms separately:

[tex]\implies 4x(4x-1)-1(4x-1)[/tex]

Factor out the common term (4x - 1):

[tex]\implies (4x-1)(4x-1)[/tex]

Apply the exponent rule:  aa = a²

[tex]\implies (4x-1)^2[/tex]

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The length of line segment AB by observation if the diagram in the task content is; 14.

It follows from the task content that the line segment MN can be considered as parallel to line segment AB. This follows from the fact that the vertices of triangle MNO are midpoints of the line segments of triangle ABC.

Consequently, it can be concluded that line segment AB is twice the length of line segment MN and hence, BC = 2 × 7 = 14.

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Answer: 380 ft and 780 ft

Step-by-step explanation:

We can assume that the flag is a rectangle. If the width of the flag is w, then the length is w + 400.

The formula for the perimeter of a rectangle is [tex]2(l+w)[/tex]. Let's plug in the values we know.

[tex]P=2(l+w)\\2320=2(w+400+w)\\1160=2w+400\\760=2w\\w=380[/tex]

We know that the length is 400 more than the width, so let's find the length.

[tex]l=w+400\\l=380+400=780[/tex]

Hence, the flag's width is 380 ft and the length is 780 ft.

32 bottles are needed to poured 4 gallons of chocolate milk.

The correct option is D.

A closed surface's volume, which is expressed as a scalar quantity, measures how much three-dimensional space is enclosed. The area that a material or 3D object takes up or contains, for instance. The SI-derived cubic metre is a common unit for quantifying volume quantitatively.

Pints to Gallons Ratio: 1:8

Permit x bottles to be present.

There are x bottles required for 4 gallons of chocolate milk.

Because of the direct proportion

4/x = 1/8

x = 4 x 8

x = 32

As a result, 32 bottles are required.

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given,

wagon weight with sand = 100 pounds

wangon weight = 80 ounces ÷ 16 = 5 pounds

sand weight = 100 - 5 = 95 pounds//

Step-by-step explanation:

3. B

because the figure to the left of the y-axis yields the one to the right when you reflect it across y = x.

4. C

Flip either of the figures vertically across x = -1 which is their intersection point and you'll get the other.

5. D

it can't be clockwise because the x value of the vertices would have been negative. so it is 270° counterclockwise resulting in ( -y , x )

6. A

Similar to #4 which except this time, you flip it horizontally like how you lay a book or pick it up.

Answer:  [tex]35 cm^2[/tex]

Step-by-step explanation:

Area of trapezium =

2X = 2 + 2 = 4 cm

2Z = 5 + 5 = 10 cm

Y = 5 cm

[tex]\dfrac{ \left( 4+10 \right) }{ 2 } \times 5[/tex]

                 |

                 |--------> 35 [tex]cm^2[/tex]

Answer:

11x9=99 16x9=144=243 144+99

The cost of the carpet required for the given rectangular floor is $1584.

A rectangle is a quadrilateral with equal pairs of sides.

Given that:

The shape of the floor is a rectangle.

The length of the floor is 16 feet.

The width of the floor is 11 feet.

So,

The area of the floor = length × breadth

                                   = 16 × 11

                                   = 176 sq feet

The area of the floor is 176 sq feet.

The cost of the carpet for a 1 sq foot floor = $9

The cost of the carpet for a 176 sq feet floor = $9 × 176

                                                                          = $1584

Thus, the cost of the carpet for a 176 sq feet floor is $1584.

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The value of angle ABT is 40 degrees if PTR is the tangent of a circle option (a) 40° is correct.

It is described as a set of points, where each point is at the same distance from a fixed point (called the center of a circle)

It is given that:

A circle is shown in the picture.

PTR is the tangent of a circle.

AT=PT

Angle APT=20°

Let O is the center of the circle:

Angle BOT = 2 Angle PAT = 40 degrees

BO = TO = r

Angle BOT = Angle TBO = (1/2)(180 - 40) = 70 degrees

RTP is the tangent

OT ⊥ PTR

Angle OTA = 90 - Angle ATR = 50 degrees

Angle ABT = 180 - Angle PAT - Angle BTA

= 180 - 20 -  (170 + 50)

= 40 degrees

Thus, the value of angle ABT is 40 degrees if PTR is the tangent of a circle option (a) 40° is correct.

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Answer:

12.8 km

Step-by-step explanation:

each mile has 1.6 km in it so in order to find the total distance in km we need to multiply the miles by 1.6= 8 miles x 1.6 =12.8 km

The vertex coefficient of the parabola with vertex (- 5, - 2) and the point (- 4, 2) is equal to 1.

In this question we must find the vertex coefficient of the equation of the parabola, which is the vertex form of the quadratic equation:

y - k = C · (x - h)²

If we know that (h, k) = (- 5, - 2) and (x, y) = (- 4, 2), then the vertex coefficient is:

2 - (- 2) = C · [- 4 - (- 2)]²

4 = C · (- 2)²

C = 1

The vertex coefficient of the parabola with vertex (- 5, - 2) and the point (- 4, 2) is equal to 1.

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The area of the parallelogram is 55 square feet

From the figure, we have the following dimensions:

Height = 5 ft

Base = 11 ft

The area of the parallelogram is calculated as:

Area = Base * Height

This gives

Area = 5ft * 11ft

Evaluate the product

Area = 55 square feet

Hence, the area of the parallelogram is 55 square feet

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Answer:

See below.

Step-by-step explanation:

Converse

The converse of a statement is formed by switching the hypothesis and the conclusion.

Given statement:  If n is an even integer, then 2n + 1 is an odd integer.

Converse:  If 2n + 1 is an odd integer, then n is an even integer.

Given statement:  If a transversal intersects two parallel lines, then each pair of corresponding angles is equal.

Converse:  If a transversal intersects two lines such that each pair of corresponding angles is equal, then the two lines are parallel to each other.

Given statement:  If each pair of opposite sides of a quadrilateral is equal, then the quadrilateral is a parallelogram.

Converse:  If a quadrilateral is a parallelogram, then each pair of opposite sides of the quadrilateral is equal.

Given statement:  If the decimal expansion of a real number is terminating, then the number is rational.

Converse:  If a real number is rational, then the decimal expansion of the number is terminating.

Answer:

x - 4

Step-by-step explanation:

monomial is one term

-11x is only one term

x-4 isn't

3 is one term

22xy is only one term

only x-4 isn't a monomial

The options A, C, E are correct. On the basis of the theory of the rotation and transformation on the parallelogram.

According to the statement

we have given that the

Parallelogram 1 is in quadrant 1 and sits on the x-axis with a point at (0, 0). Parallelogram 2 is in quadrant 4 and sits on the y-axis with a point at (0, 0)

And we have to find that the which terms are corrected from the given options when Parallelogram 1 is rotated 270 degrees counter-clockwise to form parallelogram 2.

So, For this purpose we know that the

A rotation is a type of transformation that takes each point in a figure and rotates it a certain number of degrees around a given point.

And According to the given statement

when parallelogram 1 is rotated 270 degrees counter-clockwise to form parallelogram 2.

Then the conditions A, C, E are applicable because on rotation these things are applicable.

So, The options A, C, E are correct. On the basis of the theory of the rotation and transformation on the parallelogram.

Disclaimer: This question was incomplete. Please find the full content below.

Question:

Figure 2 was constructed using figure 1.

On a coordinate plane, 2 parallelograms are shown.

Parallelogram 1 is in quadrant 1 and sits on the x-axis with a point at (0, 0). Parallelogram 2 is in quadrant 4 and sits on the y-axis with a point at (0, 0).

Parallelogram 1 is rotated 270 degrees counter-clockwise to form parallelogram 2.

For the transformation to be defined as a rotation, which statements must be true? Select three options.

A. The segment connecting the center of rotation, C, to a point on the pre-image (figure 1) is equal in length to the segment that connects the center of rotation to its corresponding point on the image (figure 2).

B. The transformation is rigid.

C. Every point on figure 1 moves through the same angle of rotation about the center of rotation, C, to create figure 2.

D.Segment CP is parallel to segment CP'.

E. If figure 1 is rotated 180° about point C, it will be mapped onto itself.

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With respect to the parent function y = cos x, the interval closest to the range of the transformed function is ( 0,50). the correct option is D.

Mathematically speaking, cosines are periodic functions, and period (T) is the distance in the horizontal axis such that f(t + T) = f(t). The period of the parent cosine function is 2π. The period in daughter functions may vary by means of the following form:

y = cos Ax     (1)

Where:

x - Independent variable

y - Dependent variable

A - Period width

From the graph, it is observed that with respect to the parent function y = cos x, the interval closest to the range of the transformed function is ( 0,50). the correct option is D.

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Answer: The answer is [0,65]

Step-by-step explanation:

Taking into account the definition of proportion, 12 students said their favorite color was blue.

The ratio is the comparison of two quantities and is measured from the division of two values. In other words, the ratio between two magnitudes is defined as their ratio or division; that is, given the quantities a and b, the ratio between them is a÷b.

The result of this division can be understood as how many times one quantity is in another.

The relationship of equality that exists between two ratios, that is, between two comparisons between two determined quantities, is known as a proportion. So if a/b corresponds to the ratio, then a/b = c/d equals a proportion.

At school, the teacher surveyed the color preferences of 21 students. Out of every 7 students surveyed, 4 said they liked blue the best. This can be expressed as a proportion as follow:

4÷7= x÷ 21

where x is the number of students of the 21 students surveyed who like blue.

Solving:

(4÷7)×21= x

12= x

Finally, 12 students said their favorite color was blue.

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The number is 841

So we can see that the number is odd.

Here we have a 3-digit number that can be written as:

a*100 + b*10 + c

Here we know that:

"the tens digit of a number is 4 times the units digit"

b = 4*c

"the hundreds digit is twice the tens digit"

a = 2*b

Now, remember that a, b, and c, are numbers on the set {0, 1, ..., 8, 9}

And if we replace the first equation into the second, we get:

a = 2*(4*c) = 8*c

Now, the only value of c that we can use such that a is a single digit whole number is c = 1

(if c = 0, then our number is zero)

a = 8*1 = 8

b = 4*1 = 4

c = 1

The number is 841

So we can see that the number is odd.

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Answer:

Let , the constant factor be x so 3 X + 5 x + 7 x = 180

Step-by-step explanation:

Find out the value of x

7x+3x+5x=180

15x=180

x=180/15

x=12

now,

     x=3 multiplied by 12=36

     x=7 multiplied by 12=84

      x=5 multiplied by 12=60

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Answer: 1208.96

Step-by-step explanation:

The volume of the cylinder is [tex](\pi)(4^2)(4)=64\pi[/tex]

The volume of the prism is [tex](6)(12)(14)=1008[/tex]

So, the total volume is [tex]1008+64\pi=\boxed{1208.96}[/tex]

Answer:

Step-by-step explanation:

x=100 bc like 180-80=100

y=80 bc vertical angles

Answer:

y = 80*

x = 100*

Step-by-step explanation:

y is the same to the 80* angle and therefore x is 100*

180* - 80 = 100*

We can infer and logically deduce that airplane’s ground speed is equal to 658.94 kph.

Given the following data:

Speed can be defined as the distance covered by an object per unit of time. Thus, speed can be measured in kilometer per hour (kph).

Mathematically, speed can be calculated by using this formula;

Speed = distance/time

In order to calculate the airplane’s ground speed, we would apply the law of cosine:

C² = A² + D² - 2(A)(D)cosθ

Substituting the given parameters into the formula, we have;

C² = 700² + 60² - 2(700)(60)cos45

C² = 434,203.03

C = √434,203.03

C = 658.94 kph.

In conclusion, we can infer and logically deduce that airplane’s ground speed is equal to 658.94 kph.

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Step-by-step explanation:

180(n-2)/n=90 ( this is the solution for the 90)

cross multiplying

180n-360=90n

180n-90n=360

90n=360

90n/90=360/90

n=4, the number of sides for the polygon with 90° is 4 sides

Solution for 100°

180(n-2)/n=100

180n-360=100n

180n-100n=360

80n=360

80n/80=360/80

n=4.5

Same applies to the rest 110°,125° ,150°,175°

The answer is A. Winter.

As the pattern is : F, W, Sp, Su, F... (letters used for convenience), then the next word in the pattern will be Winter, because that was shown in the previous sequence.

Length of the line

= [tex]\sqrt{((-2)-(-8))^{2}+(10-2)^{2} }[/tex]

= [tex]\sqrt{36+64}[/tex]

=[tex]\sqrt{100}[/tex]

= 10

x > 1 is the required solution of the given inequality - 3x + 7 < 4. This can be obtained by solving the inequality the same way as we solve an algebraic equation.

The inequality can be solved as the same way as we solve an algebraic equation.

Here in the question the inequality to be solved is given:

- 3x + 7 < 4

By subtracting 7 from both sides of the inequality we get,

⇒ - 3x + 7 - 7 < 4 - 7

⇒ - 3x < - 3

To make the side of the inequality with variable x positive we multiply the whole equation with - 1, then we have to reverse the sign of inequality since we have to multiply the whole inequality with a negative term,

we get,

⇒ 3 x > 3

Now finally by dividing the whole inequality by 3 we get,

⇒ x > 1

The number line representation can be done by marking values from 1 to numbers greater than 1.

Hence x > 1 is the required solution of the given inequality - 3x + 7 < 4.

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Answer:

[tex]e^{4x}=e^4+4e^4(x-1)+8e^4(x-1)^2+...[/tex]

[tex]\displaystyle e^{4x}=\sum^{\infty}_{n=0} \dfrac{4^ne^4}{n!}(x-1)^n[/tex]

Step-by-step explanation:

Taylor series expansions of f(x) at the point x = a

[tex]\text{f}(x)=\text{f}(a)+\text{f}\:'(a)(x-a)+\dfrac{\text{f}\:''(a)}{2!}(x-a)^2+\dfrac{\text{f}\:'''(a)}{3!}(x-a)^3+...+\dfrac{\text{f}\:^{(r)}(a)}{r!}(x-a)^r+...[/tex]

This expansion is valid only if [tex]\text{f}\:^{(n)}(a)[/tex] exists and is finite for all [tex]n \in \mathbb{N}[/tex], and for values of x for which the infinite series converges.

[tex]\textsf{Let }\text{f}(x)=e^{4x} \textsf{ and }a=1[/tex]

[tex]\text{f}(x)=\text{f}(1)+\text{f}\:'(1)(x-1)+\dfrac{\text{f}\:''(1)}{2!}(x-1)^2+...[/tex]

[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Differentiating $e^{f(x)}$}\\\\If $y=e^{f(x)}$, then $\dfrac{\text{d}y}{\text{d}x}=f\:'(x)e^{f(x)}$\\\end{minipage}}[/tex]

[tex]\text{f}(x)=e^{4x} \implies \text{f}(1)=e^4[/tex]

[tex]\text{f}\:'(x)=4e^{4x} \implies \text{f}\:'(1)=4e^4[/tex]

[tex]\text{f}\:''(x)=16e^{4x} \implies \text{f}\:''(1)=16e^4[/tex]

Substituting the values in the series expansion gives:

[tex]e^{4x}=e^4+4e^4(x-1)+\dfrac{16e^4}{2}(x-1)^2+...[/tex]

Factoring out e⁴:

[tex]e^{4x}=e^4\left[1+4(x-1)+8}(x-1)^2+...\right][/tex]

Taylor Series summation notation:

[tex]\displaystyle \text{f}(x)=\sum^{\infty}_{n=0} \dfrac{\text{f}\:^{(n)}(a)}{n!}(x-a)^n[/tex]

Therefore:

[tex]\displaystyle e^{4x}=\sum^{\infty}_{n=0} \dfrac{4^ne^4}{n!}(x-1)^n[/tex]

Matt's BAC the equation for is  mathematically given as

X= 4 standard drinks

This is further explained below.

Blood Alcohol Content, sometimes known as BAC, is a measurement that expresses the amount of alcohol present in the blood as a percentage. Because it is measured in grams per 100 milliliters of blood, a blood alcohol concentration (BAC) of 0.08 indicates that your blood contains 0.08 percent alcohol by volume.

Generally, One standard drink is equal to twelve ounces of beer.

He drank a total of 48 ounces of beer since he had three cups of beer that were each 16 ounces.

In conclusion, the number of standard drinks that Matt consumed

X= 48/12

X= 4 standard drinks

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The regression equation that represents the relationship between velocity and time is:

To find the equation, we need to insert the points (x,y) in the calculator.

For this problem, the points (t,v), that is, time and velocity, are given as follows:

(0, 6.1), (2.1, 12.2), (5.3, 17.4), (8.2, 26.4), (10, 28.1), (12.1, 32.8).

Hence, inserting these points into a calculator, the equation is given by:

v = 2.19t + 6.7.

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