A student concludes that the solution to the equation x+3=2x-4, in one variable, is the x-coordinate of the solution to the graph of this linear system.
x-y=-3
2x-y=4

Which statement justifies this conclusion based on the student's work?

A Each side of the equation, in one variable, can be found by isolating the x-variable of each equation
in the system. The solution is the y-coordinate, where the x-coordinates of both graphs are the
same.

B Each side of the equation, in one variable, can be found by isolating the y-variable of each equation
in the system. The solution is the x-coordinate, where the x- coordinates of both graphs are the
same.

C. A solution to both equations in the system is (7, 10). Since that point is where the equations
intersect, it must be the solution to the system.

D. The solution to the equation, in one variable, is x = 7. Since both equations in the system have
y-values when x = 7, it must be the solution to the system.

Therefore, the expansion of log(7x²) using the properties of logarithm is

log 7 + 2log x, expressed in terms of log x.

The mathematical idea of a logarithm illustrates the opposite of a power. It follows that the exponential that must be multiplied by bc to get the value of a number, x, is equal to its ground logarithm. As an illustration, 1000 = 103, so log10 = 3 is 1000's base-10 logarithm, which is 3. As an example, the square of ten is now only one twentieth of the base-10 exponential of ten, which is two. Log 100 = 2. Mathematicians use the logistic (or log) notion to respond to problems like these.

Here,

Using the product rule of logarithms, we can write:

log(7x²) = log 7 + log(x²)

Using the power rule of logarithms, we can write:

log(7x²) = log 7 + log(x²) = log 7 + 2log x

Therefore, the expansion of log(7x²) using the properties of logs is:

log 7 + 2log x, expressed in terms of log x.

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The optimal Bayesian classifier is the one that minimizes the probability of misclassification. This can be achieved by choosing the class that has the highest posterior probability given the observed data. In this case, we need to compare the conditional class probabilities P2(t) and P0(I) for each data point and choose the class with the higher probability.

To find the optimal Bayesian classifier, we need to find the values of a and b that define the decision boundary between the two classes. This can be done by setting P2(t) = P0(I) and solving for x:

P2(t) = P0(I)
Pr(Y; = 1|X, = x) = Pr(Y; = 0[Xi = 1)
1 - P2(I) = P2(I)
2P2(I) = 1
P2(I) = 0.5

Now we can solve for x to find the decision boundary:

0.5 = Pr(Y; = 1|X, = x)
0.5 = P2(I)
x = a + b

Therefore, the optimal Bayesian classifier is of the form:

if Xe [a, b] then yi = 1
if Xe [b, a] then yi = 0

where a and b are the values of x that satisfy P2(I) = 0.5.

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The volume of the triangular pyramid shown  is (c) 104 cubic units

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

X = 6 units, Y = 4 units, and Z = 13 units

The volume is calculated as

Volume = xyz/3

Substitute the known values in the above equation, so, we have the following representation

Volume = 6 * 4 * 13/3

Evaluate

Volume = 104

Hence, the volume is (c) 104 cubic units

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Answer:

P = 15 + 24y

Step-by-step explanation:

The perimeter of a rectangle is the sum of the lengths of all four sides. If one side of the rectangle has a length of 7.5 and the other side has a length of 12y, we can write the expression for the perimeter as:

P = 2(7.5) + 2(12y)

Simplifying this expression, we get:

P = 15 + 24y

Therefore, the simplified expression for the perimeter of the rectangle with one side of length 7.5 and the other side of length 12y is 15 + 24y.

When a cube's top is chopped off to create a rectangular box, the original cube's side length is 5.43 feet as a result.

A rectangular is a square with four right angles in Euclidean geometry and trigonometry. You may also speak to it as an equiangular tetra, meaning that all of its edges are equal. A straight angle could also be found in the parallelogram. Rectangles with four evenly sized sides are called squares. Four 90 ° vertices and equal equal sides make up a quadrant with the look of a rectangle. As a basis, it's also known as a raster images rectangle. A rectangle is also referred to as a trapezoidal since its opposite angles are equal but parallel.

given

Let x represent the cube's edge in feet.

Cube volume equals side 3

Hence, the given cube's volume,

The top of a cube is now chopped off in a 4 foot thick chunk.

The slice's volume is equal to x x x 4 = 4x2 ft2.

The volume of the final figure is therefore equal to the sum of the volumes of the cut and cube figures.

= ( x³ - 4x² ) ft²

The answer to the query is

Finding the zeros of the function will lead to the discovery of the solution to the aforementioned equation ( by graphing ),

We discovered that

The equation's graph crosses x at (5.426,0)

Hence, 5.426 is the equation's zero.

⇒ x = 5.426 ≈ 5.43

When a cube's top is chopped off to create a rectangular box, the original cube's side length is 5.43 feet as a result.

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The probability that at least one of the students will select a president who did not serve in the military is 1.00.

The probability that a student selects a president who did serve in the military is:

P(selects military president) = 31/43

Therefore, the probability that a student selects a president who did not serve in the military is:

P(selects non-military president) = 1 - 31/43 = 12/43

The probability that none of the 888 students select a non-military president is:

P(none select non-military president) = (31/43)^{888}

Therefore, the probability that at least one of the students will select a president who did not serve in the military is:

P(\text{at least one not in military}) = 1 - P(none select non-military president) = 1 - (31/43)^{888} ≈ 0.9996

Rounded to the nearest hundredth, the probability is 1.00.

What is probability?

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event. For example, the probability of flipping a fair coin and it landing on heads is 0.5 or 50%, while the probability of rolling a 7 on a standard six-sided die is 6/36 or 1/6, since there are six ways to roll a 7 out of 36 possible outcomes. Probability theory is used in many fields, including statistics, mathematics, science, finance, and engineering, to model and analyze random events and to make predictions based on data.

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Answer:

Confidence interval = -0.1013 ± 0.1991

Confidence interval = (-0.30, 0.10)

Step-by-step explanation:

We can use the following formula to find the confidence interval for the difference in two population proportions:

Confidence interval = (p1 - p2) ± z*sqrt[p1(1-p1)/n1 + p2(1-p2)/n2]

where p1 and p2 are the sample proportions, n1 and n2 are the sample sizes, and z is the z-score corresponding to the level of confidence.

First, we need to calculate the sample proportions:

p1 = 104/252 = 0.4127

p2 = 55/107 = 0.5140

Next, we need to find the value of z for a 99% confidence level. We can look up this value in a standard normal distribution table or use a calculator, which gives us a z-score of 2.576.

Then, we can plug in the values:

Confidence interval = (0.4127 - 0.5140) ± 2.576*sqrt[0.4127(1-0.4127)/252 + 0.5140(1-0.5140)/107]

Simplifying this expression, we get:

Confidence interval = -0.1013 ± 0.1991

Rounding to two decimal places and using interval notation, we get:

Confidence interval = (-0.30, 0.10)

The angles which are co-terminal with the given angle and less than 1080 but greater than -720 are 450°, 810°, -270° and -630°. The solution has been obtained using concept of co-terminal angles.

What is co-terminal angle?

Co-terminal angles are those with a common terminal side that are in standard position (angles with the starting side on the positive x-axis). The angles where the difference could be 360 degrees or multiples of 360 degrees are known as co-terminal angles.

The calculation of the two positive angles and the two negative angles, which are co-terminal, is as follows:

(a) For 90°, the two positive angles are

90° + 360° = 450°

450° + 360° = 810°

(b) The two negative angles are

90° - 360° = -270°

-270° - 360° = -630°

Hence, the angles which are co-terminal with the given angle and less than 1080 but greater than -720 are 450°, 810°, -270° and -630°.

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Answer:

Step-by-step explanation:

75 x 2 = 150mph

150 x 8 = 1200 miles

The train will travel 1200 miles in 8 hours.

Speed is defined as the ratio of the time distance traveled by the body to the time taken by the body to cover the distance. Speed is the ratio of the distance traveled by time. The unit of speed in miles per hour.

If the train travels 75 miles in 1/2 hour, we can use this information to find the train's speed:

Speed = Distance/Time

Speed = 75 miles / 0.5 hour

Speed = 150 miles/hour

Now that we know the speed of the train, we can use it to find how many miles the train will travel in 8 hours:

Distance = Speed x Time

Distance = 150 miles/hour x 8 hours

Distance = 1200 miles

Therefore, the train will travel 1200 miles in 8 hours.

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The answer is the expected value is - 2. The equation for expected value is the sum of the values times their probabilities.

The total length of the stickers that are more than 2 inches long is 5 1/2 inches. The Option C is correct

A length is the measurement or extent of something from end to end; the greater of two or the greatest of three dimensions of an object.

The calculation of the total length of the stickers that are more than long is as follows:

= 2 1/4 + 2 1/2 + 2 3/4 + 3

= 9/4 + 5/2 + 11/4 + 3

= 9 + 10+ 11 + 12 / 4

= 42/4

= 10 2/4 inches

= 5 1/2 inches

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Full question "Jalesia has a set of train stickers. She measures the length of the stickers. Then she makes a line plot of the data. h of Stickers In jalesia takes all the stickers that are more than  long. She lays them end to end across her folder. What is the total length of the stickers that are more than  long?"

The completed blanks in Nyala's solution that is used to prove that the measure of angle x is 40° is as follows;

If we perform a 180° rotation about the point O the following happens

Ray [tex]\overleftrightarrow{EF}[/tex] maps unto ray [tex]\overleftrightarrow{GH}[/tex]

Ray [tex]\overleftrightarrow{GH}[/tex] maps unto ray [tex]\overleftrightarrow{FE}[/tex]x

[tex]\overleftrightarrow{FG}[/tex] maps unto itself

Therefore, the image of angle x will be exactly where the pre-image of 40° was. Since rotation preserve angle measure, the measure of angle x must be 40°.

The measure of an angle is the degree of rotation between the initial side and the terminal side of the angle.

The details of the method Nyala used to prove that the measure of the angle x is 40° can be found as follows;

The rotation of a line 180° about the center of the line maps onto itself, and the rotation of a line about a non colinear point, is parallel to the original line.

Whereby the center of rotation of the line [tex]\overleftrightarrow{EF}[/tex] is equidistant from both the line [tex]\overleftrightarrow{GH}[/tex] which is parallel to the line [tex]\overleftrightarrow{EF}[/tex], the image of the line [tex]\overleftrightarrow{EF}[/tex] maps to the line [tex]\overleftrightarrow{GH}[/tex] following a 180° rotation about the point O, and the rotation of the line [tex]\overleftrightarrow{FG}[/tex] about the point O maps unto itself, such that the angle x maps to the angle 40°, therefore;

∠x ≅ 40° (A rotation transformation is a rigid transformation)

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The experimental probability that the next student to respond likes Hamlet best is 1/6 or 0.1667 as a decimal.

In this problem, we are given the results of a survey conducted in an English class to determine which Shakespeare play the students liked the best. The teacher recorded the number of students who liked each of the three plays: Hamlet, Twelfth Night, and A Midsummer Night's Dream.

To calculate the experimental probability, we first need to find the total number of students who have responded to the survey so far. From the information given, we know that 2 students liked Hamlet best, 8 students liked Twelfth Night best, and 2 students liked A Midsummer Night's Dream best. Therefore, the total number of students who have responded so far is: 2 + 8 + 2 = 12

Next, we need to determine how many of these 12 students liked Hamlet best. We know from the data given that 2 students liked Hamlet best, so the number of students who liked Hamlet best is: 2

Finally, we can calculate the experimental probability that the next student to respond likes Hamlet best. We do this by dividing the number of students who liked Hamlet best (2) by the total number of students who have responded so far (12): 2/12

Simplifying this fraction, we get: 1/6

Therefore, the experimental probability that the next student to respond likes Hamlet best is 1/6 or 0.1667 as a decimal. This means that if we were to randomly select a student from the class and ask them which Shakespeare play they liked the best, there is a 1 in 6 chance that they would choose Hamlet.

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The amplitude, period and frequency are 1, 2π and 1/2π

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

f(x) = sin(x - π/2) - 2

A sinusoidal function is represented as

f(x) = Asin(B(x + C)) + D

Where

Amplitude = A

Period = 2π/B

So, we have

A = 1

Period = 2π/1

Period = 2π

For the frequency, we have

frequency = 1/Period

frequency  = 1/2π

Hence, the frequency is 1/2π

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Are you more likely to see a twelve-pack from the local brewery or a twelve-pack from the large chain with an average fill of 327.9 ml?

Local brewery.

a. At the local brewery the middle 95% of six-packs will have an average fill between ml 326.828 and 329.572 ml

b. At the large chain brewery, the middle 95% of six-packs will have an average fill between 326.824 ml and 329.176 ml.

X ζ N (328.2, 0.7^2)

YζN (328, 0.6^2)

P (X> 327.9) = 0.6659

P (Y> 327.9) = 0.5662

Since, P (X> 327.9) > P (Y> 327.9)

We are more likely to see 12-pack from the local brewery.

a. To find a, b such that P(a<x<b) = 0.95

a= 326.828

b= 329.572

b. To find c, d such that P(c<Y<d) = 0.95

c= 326.824

d= 329.176

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Answer:

£2,700

Step-by-step explanation:

Ryan pays a deposit for the sofa then pays the rest of the cost in 24 equal payments of £112.50 b) How much was the deposit?

If Ryan paid the rest of the cost in 24 equal payments of £112.50, then the deposit would have been the total cost minus the total amount of the 24 payments, which is £2,700. Therefore, the deposit was £2,700.

Answer:

Step-by-step explanation:

15% of a price of 116 dollars would be a price decrease of $17.40. This means that the price when on sale would be $98.60 as you have to subtract 17.40 from 116.

The value of the function [tex]\lim_{x \to \ -1} \frac{2}{(x + 1)^{2}}[/tex] is ∞.

A value of the function, whenever the input values are substituted in the function then the function approaches some number.

And that number is a limit of the function.

Given:

A function,

f(x) = 2/{(x + 1)²}

And function increases without bounds as the values of x approach -1.

So,

[tex]\lim_{x \to \ -1} \frac{2}{(x + 1)^{2}}[/tex]

= 2/(x² + 2x + 1)

= 2/{1 + (-2) + 1}

= 2/0

= ∞

Therefore, the function approaches infinity.

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Answer:

Sophie's pyramid is approximately 4.8 inches.

Step-by-step explanation:

The formula for the volume of a square pyramid is given by:

V = (1/3) * B * h

where V is the volume, B is the area of the base, and h is the height of the pyramid.

We are given that the volume of Sophie's pyramid is 90 cubic inches, and the base has a side length of 7.5 inches. The area of the base is:

B = (7.5)^2 = 56.25 square inches

Substituting these values into the formula for the volume, we get:

90 = (1/3) * 56.25 * h

Multiplying both sides by 3, we get:

270 = 56.25 * h

Dividing both sides by 56.25, we get:

h = 4.8 inches (rounded to one decimal place)

Therefore, the height of Sophie's pyramid is approximately 4.8 inches.

Answer:

The volume of the rectangular prism is 96.

Step-by-step explanation:

Use the formula L × W × H to get your answer. Plug in the givem measurements. 8 × 4 × 3 = 96.

With a weekly budget of $12, Mark will need to save money for 9 weeks, 5 days in order to purchase the guitar.

The value is said to be in its simplified form when divided through its smallest equivalent fraction. How to find the most basic form. Find common factors in the denominator and numerators. Verify a minority integer t verify if it qualifies like a prime number. A statement comprising two rotational symmetry and an equal sign in the middle is called an equation in mathematics. The general form of almost any equation contains the degrees of all variables in descending order. The conventional form of something like a linear equation is an x + b = 0. The general form of a mathematical expression with two variables is an x + b y + c = 0. (or something similar). Equations can be categorized as identities or conditional solutions. An identity holds true irrespective of the value given to the variables.

given

let the week be x

equation =  $55 + 12x = $169.00

55 + 12x = 169

12x = 169 - 55

12x = 114

x = 9.5

With a weekly budget of $12, Mark will need to save money for 9 weeks, 5 days in order to purchase the guitar.

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The Definition of the Derivative correctly stated by

In mathematics, a derivative is the rate at which a function changes in relation to a variable. Calculus and differential equations issues must be solved using derivatives.

As, we know by definition of derivative

= [tex]\lim_{h \to 0}[/tex] f(x+h)- f(x) / h

Also, the Derivative also shows the Slope of the secant line.

and, derivative is widely used to the rate of change then it also shows

Instantaneous rate of change.

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The possible lengths of side AB as required to be determined in the task content are; 17.2, 15.59 and 8.

It follows from the task content that the length measures which are possible for side AB be determined.

By the triangle inequality theorem;

AB < 7 + 12; AB < 19

and

AB > 12 - 7; AB > 5

On this note, 5 < AB < 19.

Consequently, the possible lengths of side AB as required are; 17.2, 15.59 and 8.

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The two equations that have the same solution as the original equation are: 23p - 101 = 65p - 40 - p, 2.3p - 14.1 = 6.4p - 4

An equation is a mathematical statement that shows that two expressions are equal. It is usually written as an expression on the left-hand side (LHS) and an expression on the right-hand side (RHS) separated by an equal sign (=).

The expressions on both sides of the equation can contain variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. The variables in an equation represent unknown values that can vary, while the constants are fixed values that do not change.

Equations are used to represent mathematical relationships or describe real-world situations. They can be used to solve problems, make predictions, and test hypotheses.

To solve an equation, one must find the value of the variable that makes the LHS equal to the RHS. This is done by performing mathematical operations on both sides of the equation to isolate the variable. The goal is to get the variable by itself on one side of the equation, with a specific value on the other side.

Equations can be simple or complex, linear or nonlinear, and can involve one or more variables. Examples of equations include:

2x + 5 = 13

y = 3x^2 - 2x + 7

4a + 2b - 3c = 10

Equations are used in many areas of mathematics and science, including physics, chemistry, and engineering, among others.

The equations that have the same solution as 2.3p – 10.1 = 6.5p – 4 – 0.01p are:

23p – 101 = 65p – 40 – p

2.3p – 14.1 = 6.4p – 4

To rewrite the given equation in a different form, we can simplify it using the distributive property and combining like terms:

2.3p - 10.1 = 6.5p - 4 - 0.01p

2.3p - 6.5p + 0.01p = -4 + 10.1

-4.19p = 6.1

p = -1.456

Now we can plug this value of p into each of the answer choices and see which ones give the same result:

23p - 101 = 65p - 40 - p

24p - 101 = 64p - 40

p = 3.5

This solution is not the same as the solution we found for the original equation, so this choice is not correct.

2.3p - 14.1 = 6.4p - 4

8.7p = 10.1

p = 1.16

This solution is also not the same as the solution we found for the original equation, so this choice is not correct.

Therefore, the two equations that have the same solution as the original equation are:

23p - 101 = 65p - 40 - p

2.3p - 14.1 = 6.4p - 4

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The length of the 180° arc as required in the task content is; 22/21 inches.

Recall that the length of an arc is given by the formula;

Length of arc = theta / 360 × 2 π r

Therefore, for a radius of 3 inches;

Length of arc = 180/360 × 2 × 22/7 × 3

Length of arc = 22/21 inches.

Ultimately, the length of the arc in discuss is; 22/21 inches.

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The correct statement is:

E) There is exactly one solution at (1, -2).

What is equation ?

An equation is a mathematical statement that indicates that two expressions are equal. It contains variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. An equation typically consists of two sides separated by an equal sign (=). The values of the variables that make the equation true are called solutions or roots of the equation. Equations are used to model a wide range of phenomena in mathematics, science, engineering, and other fields, and they play a central role in problem-solving and mathematical analysis.

According to given conditions:

To solve the system of equations:

y = 4x - 6 ...(1)

x + 3y = -5 ...(2)

We can use substitution or elimination method. Here, we will use the substitution method:

From equation (1), we have y = 4x - 6.

Substitute y = 4x - 6 in equation (2):

x + 3(4x - 6) = -5

Simplifying the above equation, we get:

13x - 18 = -5

Adding 18 on both sides, we get:

13x = 13

Dividing both sides by 13, we get:

x = 1

Substitute x = 1 in equation (1):

y = 4(1) - 6 = -2

Therefore, the solution to the system of equations is (1, -2).

So, the correct statement is:

A) There is exactly one solution at (0, -6). (Incorrect)

B) There is exactly one solution at (12). (Incorrect)

C) There is no solution. (Incorrect)

D) There are infinitely many solutions. (Incorrect)

The correct statement is:

E) There is exactly one solution at (1, -2).

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Answer:

No solution

Step-by-step explanation:

We are given the following equation and asked to solve for x

[tex]\dfrac{\left(x+2\right)}{2x-2}=\dfrac{1}{2}[/tex]

[tex]\mathrm{Apply\:fraction\:cross\:multiply:\:if\:}\dfrac{a}{b}=\dfrac{c}{d}\mathrm{\:then\:}a\cdot \:d=b\cdot[/tex]

[tex]\left(x+2\right)\cdot \:2=\left(2x-2\right)\cdot \:1[/tex]

[tex]\mathrm{Simplify}2x + 2 = 2x - 2\\\\\textrm{Subtract\;$2x$\;both\;sides}:\\-2 = 2[/tex]

Since the sides are not equal this indicates there is no solution to this system.

Answer:

Step-by-step explanation:

We can use the law of sines to find the length of side a. The law of sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle. In other words:

a/sin A = b/sin B = c/sin C

where a, b, and c are the lengths of the sides of the triangle, and A, B, and C are the angles opposite those sides.

We are given that A = 80°, B = 60°, and b = 15 cm. We can substitute these values into the law of sines and solve for a as follows:

a/sin 80° = 15/sin 60°

Multiplying both sides by sin 80°, we get:

a = 15*sin 80°/sin 60°

Using a calculator, we can evaluate sin 80° and sin 60° as follows:

sin 80° ≈ 0.9848

sin 60° = √3/2 ≈ 0.8660

Substituting these values, we get:

a ≈ 15*0.9848/0.8660

a ≈ 17.06

Rounding this to the nearest whole number, we get:

a ≈ 17

Therefore, the length of side a is approximately 17 cm.