Solve the system of equations by transforming a matrix representing the system of equation into reduced row echelon form.

2x + y − 3z= −20
x + 2y + z= −3
x − y + 5z= 19

What is the solution to the system of equations?

Answer:

sorry  can you write it down and take a pic of it

please that is the only way i can answer it

Step-by-step explanation:

Step-by-step explanation:

t=2+5×j

j=5

t=2+5×5

t=27

Tom has 2 more than 5 times what Jane has, so you would multiply the amount Jane has by 5, then add 2 to that:

Tom  = 5(5) +2

Tom = 25 +2

Tom = 27 CD's.

Answer: 12/2652 or 1/221

Step-by-step explanation:

There are 4 kings in a deck

So the probability of getting a king would be

4/52 then after receiving a king and not replacing you will the have a 3/51 chance

So all together you will have a:

4/52 * 3/51 = 12/2652 or simplified 1/221

Hope this helps

The probability of drawing a King from a deck of 52 cards without replacement and then drawing a second King is 1/221.

We have,

To find the probability of drawing a King from a deck of 52 cards without replacement, and then drawing a second King, we can calculate it as follows:

The probability of drawing a King as the first card is 4/52 since there are 4 Kings in a deck of 52 cards.

After removing one King from the deck, there are now 51 cards left, including 3 Kings.

The probability of drawing a second King, given that a King has already been drawn, is 3/51.

To find the overall probability of both events occurring, we multiply the individual probabilities:

(4/52) * (3/51) = 12/2652

Simplifying the fraction, we have:

12/2652 = 1/221

Therefore,

The probability of drawing a King from a deck of 52 cards without replacement and then drawing a second King is 1/221.

Learn more about probability here:

https://brainly.com/question/14099682

#SPJ2

Answer:

The sampling technique used here is Cluster Sampling.

Step-by-step explanation:

Since here Population is divided into different parts called grid and whole elements of some selected grid is taken as sample. So, Cluster Sampling is used here.

Further the different types of sampling we have are:

Simple Random Sampling is the sampling where samples are chosen randomly, where each unit has an equal chance of being selected in a sample.

If the population is divided into a different group called cluster and all elements of clusters are selected as a sample then it is Cluster Sampling.

In Convenience sampling, observers collect the sample as his\her convenience.

In Systematic Sampling sample is chosen by some criteria like he\she is taken every 10th unit as a sample from the population.

In Stratified Sampling population is divided into several groups such that within the group it is homogeneous and between the group it is heterogeneous. And now a selection of each stratum and unit has an equal chance of selection.

Answer:

  $64,720

Step-by-step explanation:

The straightforward way to figure this is ...

  total pay = straight commission + annual bonus

  = 1.5% × 2,555,500 + 2.5% ×(2,555,500 -1,500,000)

However, this expression can be simplified a little bit to ...

  2,555,500 × (1.5% +2.5%)  -2.5% × 1,500,000

  = 4% × 2,555,000 -2.5% × 1,500,000

  = 102,220 -37,500

  total pay = $64,720

Answer:

The correct graph is the second one, that the line intersects x at 6 and y at 11.5

Step-by-step explanation:

Samuel has to sell at least $90. So, in this graph if he sell only child ticket, he will have to sell 11.5 tickets. Or if he sell only adult tickets, he will have to sell at least 6.

Answer:

The last graph is the best models this situation.

Step-by-step explanation:

First we need to find the equation of ticket selling. To not loss any money from this business Samuel need to sell at least 6 adult or 11.25 child tickets. I know ticket number must be integer but those numbers are x and y values that line crosses through axes. The equation is:

[tex]8x+15y\geq 90[/tex]

and the graph of this equation is attached.

Answer:

grams of gold=8x10^12 g

Step-by-step explanation:

first we calculate the volume of the ocean this is achieved by multiplying the area by the depth

A=3.63x10^8  km^2=3.63x10^14   m^2

V=AxL

V=3.63x10^14 x 3800=1.3797x10^18m^3=1.3797x10^21L

then we multiply the volume in liters by the concentration of gold in the ocean per liter

grams of gold=1.3797x10^21   x    5.8x10^-9=8x10^12 g

Answer:

8.00 *10¹² g

Step-by-step explanation:

We must calculate the grams of gold in the ocean.

We are given the concentration of dissolved gold as grams per Liter.

So we need to first calculate the Liters of the ocean, that is, the volume.

We can calculate the volume of the ocean assuming a prism shape as:

Volume = Area x Depth

We should first convert the area from km² to m² so that the units are consistent:

[tex]3.63 *10^{8}  km^{2} * (\frac{1000 m}{1 km} )^{2} = 3.63 *10^{14}  m^{2}[/tex]

So the volume:

Volume = 3.63 x10¹⁴ m² * 3800 m = 1.38 x10¹⁸ m³

Since we have the concentration given in Liters, lets convert the volume to Liters, knowing that 1 L = 1000 m³:

[tex]1.38 *10^{18} m^{3} *\frac{1000 L}{1 m^{3} } = 1.38 *10^{21} L[/tex]

Knowing that the concentration of gold is 5.8 *10⁻⁹ grams per Liter, we can multiply this value by the Liters of the ocean to calculate the grams of gold:

5.8 *10⁻⁹ g/L  x 1.38 *10²¹ L = 8.00 *10¹² g

There are 8.00 *10¹² grams of gold in the ocean

Answer:

y = -1½x + 3 11⁄14

Step-by-step explanation:

First convert from Standard Form to Slope-Intercept Form:

2x - 3y = 4

-2x - 2x

____________

-3y = -2x + 4

___ _______

-3 -3

y = ⅔x - 1⅓ >> Slope-Intercept Form

↑

slope

Now, Perpendicular Lines have OPPOSITE MULTIPLICATIVE INVERSE Rate of Changes [Slopes], so since the slope is ⅔, the opposite multiplicative inverse of that would be -1½, or -3⁄2. Anyway, do the following:

4 = -1½[-⅐] + b

3⁄14

-3⁄14 - 3⁄14

_______________

3 11⁄14 = b

y = -1½x + 3 11⁄14 >> New equation

I am joyous to assist you anytime.

Answer:

Step-by-step explanation:

So a few things to know before hand.  Slope ntercept form is y = mx + b where m is the slope, and in this form that will always make b the y intercept.

A perpendicular slope is pretty easy to find.  Of course, perpendicular means it intersects that first line at a 90 degree angle.  So the x and y axis themselves are perpendicular.  Anyway, if you know the slope of the line, a perpendicular slope is -1/m where m is the slope.  S taking the simplest example, in the graph of just x, the slope is 1, so the perpendicular slope is -1/1 or just -1.

The last thing is to know how to write the equation of a linear function when you know a point and its slope.  if you have the slope m and a point on the graph (a,c) you can use the point slope form which is this.  y - c = m(x - a) where you solve for y.  x an y stay as variables here.

Now knowing all that we can start.  First we want to put the original graph into slope intercept form, which is pretty easy.  Just manipulate the equation.

2x - 3y = 4

2x -4 = 3y

y = (2x - 4)/3

y = 2/3 x - 4/3

so m = 2/3 and b = -4/3

Now, we have the slope of this line and want the slope of a perpendicular line.  Like I mentioned before the slope is -1/m so in this case that's -1/(2/3) = -3/2  Let me know if you don't get how that was gotten.

Now that we kno the perpendicular slope, we can make a perpendicular line.  How do we make a line when we know the slope and a point?  Keep in mind the point is (-1/7, 4)

y - c = m(x-a)

y - 4 = -3/2(x + 1/7)

y - 4 = -3/2 x - 3/14

y = -3/2 x + 53/14

If it weren't in slope intercept form you'd have to put it in that, but I took care of it in the process, so here's the answer.  Let me know if there's anything you don't understand.  

Answer:

see explanation

Step-by-step explanation:

Inequalities of the type | x | > a, always have solutions of the form

x < - a or x > a

This can be extended to expressions, that is

14 - 5x < - 8 OR 14 - 5x > 8 ( subtract 14 from both sides of both inequalities )

- 5x < - 22 OR - 5x > - 6

Divide both sides by - 5 , reversing the inequality sign as a consequence

x > [tex]\frac{22}{5}[/tex] OR x < [tex]\frac{6}{5}[/tex]

That is the solution is

x < [tex]\frac{6}{5}[/tex] OR x > [tex]\frac{22}{5}[/tex]

Answer:

Step-by-step explanation:

Inequalities of the type | x | > a, always have solutions of the form

x < - a or x > a

This can be extended to expressions, that is

14 - 5x < - 8 OR 14 - 5x > 8 ( subtract 14 from both sides of both inequalities )

- 5x < - 22 OR - 5x > - 6

Divide both sides by - 5 , reversing the inequality sign as a consequence

x >  OR x <

That is the solution is

x <  OR x >

Read more on Brainly.com - https://brainly.com/question/13211758#readmore

Answer:

Rachel's score is 2.3333 standard deviations above the mean

Step-by-step explanation:

GRE scores are normally distributed

Let be G the random variable ''Gre scores''

G  ~ N (mean,standard deviation)

G ~ N (600,30)

Rachel scored 670 on the analytic portion of the GRE.

670 - 600 will be the score above the mean

670 - 600 = 70

To find this in terms of standard deviation we divide by the standard deviation

70/standard deviation = 70/30 = 7/3 = 2.33333333 standard deviations

Rachel's GRE score is approximately 2.33 standard deviations above the mean. The calculation is made by subtracting the mean from the observed score and dividing this by the standard deviation.

The subject of this question pertains to the mathematical concept of Z-scores, used in statistics to measure how many standard deviations an element is from the mean. In the case of Rachel's GRE score, we can calculate the number of standard deviations her score is above the mean using the formula z = (X - μ) / σ where:

By substituting these values into the formula, we get:

z = (670 - 600) / 30 = 70 / 30 = 2.33

This means that Rachel's score is approximately 2.33 standard deviations above the mean.

https://brainly.com/question/15016913

#SPJ3

Answer:

Part a) 9*5+2

Part b) 7*6+4

Part c) Lena is correct

Part d) see the explanation

Step-by-step explanation:

The complete question in the attached figure

Part a) Write a numerical expression to represent Lena’s way of counting

To represent Lena’s way of counting, multiply the number of stacks by the number of pennies in each stack plus the number of pennies left over

Let

x -----> the number of stacks

y ----> the number of pennies in each stack

z ----> the number of pennies left over

so

[tex]xy+z[/tex]

we have

x=9 stacks

y=5 pennies

z=2 pennies

substitute

[tex]9*5+2[/tex]

Part b) Write a numerical expression to represent her mother's way

To represent her mother’s way of counting, multiply the number of stacks by the number of pennies in each stack plus the number of pennies left over

Let

x -----> the number of stacks

y ----> the number of pennies in each stack

z ----> the number of pennies left over

so

[tex]xy+z[/tex]

we have

x=7 stacks

y=6 pennies

z=4 pennies

substitute

[tex]7*6+4[/tex]

Part c) Lena thinks her mother must have been working with fewer pennies than she was.  Is Lena correct? 

we have that

Lena’s expression

[tex]9*5+2[/tex]

Simplify

[tex]9*5+2=47[/tex]

Her mother’s expression

[tex]7*6+4[/tex]

Simplify

[tex]7*6+4=46[/tex]

therefore

Lena’s expression is more.

Lena is correct

Part d) Use a  < ,  > , or  =  symbol to show how the two expressions compare

[tex]9*5+2 > 7*6+4[/tex]

[tex]47 > 46[/tex]

The number 47 is greater than the number 46

therefore

The symbol is  " >"

Answer:

144 malt balls will fit in the carton

Step-by-step explanation:

* Lets explain how to solve the problem

- To solve the problem we must to know each dimensions of the

 cartoon will fit how many balls

- To do that divide each dimension by the diameter of the ball

∵ The diameter of the chocolate malt ball is 0.75 inches

∵ The dimensions of the carton are 3 inches , 3 inches , 7 inches

* Lets find how many balls will fit in the side of 3 inches

∵ 3 ÷ 0.75 = 4

∴ There are 4 balls will fit in the side of 3 inches

∵ Two dimensions of the carton are 3 inches

∴ There 4 × 4 balls fit in the base of the carton

∵ The height of the carton is 7

* Lets find how many balls can fit in the height

∵ 7 ÷ 0.75 = 9.3333

∴ 9 balls can fit the height of the carton

∴ There are 4 × 4 × 9 balls will fit in the carton

∴ The number of the balls = 4 × 4 × 9 = 144 balls

* 144 malt balls will fit in the carton

Explanation:

The term containing the variable, e^-t has a range from 0 to infinity, as all exponential terms do.

For t → -∞, e^-t → ∞ and the value of the rational expression becomes 5/∞ ≈ 0. That is, there is a horizontal asymptote at f(t)=0 for large negative values of t.

For t → ∞, e^-t → 0 and the value of the rational expression becomes approximately 5/2. That is, there is a horizontal asymptote at f(t) = 5/2 for large positive values of t.

Essentially, the curve is "S" shaped, with a smooth transition between 0 and 5/2 for values of t that make 3e^-t have values within an order of magnitude of the other term in the denominator, 2.

At t=0, 3e^-t = 1 and the denominator is 2+3=5. That is, f(0) = 5/5 = 1. Of course, the curve will cross the line f(t) = 5/4 (halfway between the asymptotes) when 3e^-t = 2, or t=ln(3/2)≈0.405. The curve is symmetrical about that point.

You can sketch the graph by finding values of t that give you points on the transition. Typically, you would choose t such that 3e^-t will be some fraction or multiple of 2, say 1/10, 1/3, 1/2, 1, 2, 3, 10 times 2.

___

f(t) is called a "logistic function." It models a situation where growth rate is proportional both to population size and the difference between population size and carrying capacity. In public health terms, it models the spread of disease when that is proportional to the number of people exposed and to the number not yet exposed.

Answer:

10C5=252

6C5=6

6C5/10C5= 1/42

Answer: we can use the folowing polynomial.

P(x) = [tex]\frac{y1 (x - x2)(x -x3)}{(x1 - x2)(x1-x3)}[/tex] + [tex]\frac{y2 (x - x1)(x -x3)}{(x2 - x1)(x1-x3)}[/tex] + [tex]\frac{y3 (x - x2)(x -x1)}{(x3 - x2)(x3-x1)}[/tex]

you can see that P(x1) = y1

                          P(x2) = y2

                          P(x3) = y3

this is a Lagrange polynomial.

Answer:

Lee has more pizza

Lee has 2.24 in^2 more than John

Step-by-step explanation:

step 1

Find the area of each slice of pizza

[tex]A=\frac{1}{8}\pi r^{2}[/tex]

we have

[tex]r=16/2=8\ in[/tex] ----> the radius is half the diameter

substitute

[tex]A=\frac{1}{8}\pi 8^{2}[/tex]

[tex]A=8\pi\ in^{2}[/tex]

step 2

Find the area of John's part (area of shaded triangle)

The measure of the central angle of each slice of pizza is equal to

[tex]360\°/8=45\°[/tex]

so

the height of triangle is equal to the base

Let

x ---->the base of the shaded triangle

[tex]cos(45\°)=\frac{x}{r}[/tex]

[tex]cos(45\°)=\frac{x}{8}[/tex]

Remember that

[tex]cos(45\°)=\frac{\sqrt{2}}{2}[/tex]

substitute

[tex]\frac{\sqrt{2}}{2}=\frac{x}{8}[/tex]

solve for x

[tex]x=4\sqrt{2}\ in[/tex]

Find the area of shaded triangle

[tex]A=(1/2)(4\sqrt{2})(4\sqrt{2})=16\ in^2[/tex]

step 3

Find the area of Lee's part

The area of Lee's part is equal to the area of two slices of pizza minus the area of two triangles

so

[tex]2(8\pi)-2(16)=(16\pi-32)\ in^2[/tex]

assume

[tex]\pi =3.14[/tex]

[tex](16(3.14)-32)=18.24\ in^2[/tex]

so

Lee's part is greater than John part's

Find the difference

[tex]18.24-16=2.24\ in^2[/tex]

therefore

Lee has more pizza

Lee has 2.24 in^2 more than John

Without exact dimensions of Lee's cut, we can't calculate the precise difference in area after he cuts off an edge, but John likely has more pizza since his slice is unmodified.

The student's question involves comparing areas of pizza slices after one has been modified by cutting off an edge. To answer who has more pizza and by how much, we need to calculate the area of the pizza slices. The original pizza is 16 inches in diameter, and when divided into 8 equal slices, each slice is a sector of a circle with a central angle of 45 degrees. If Lee cuts off an edge and John takes an unmodified triangular slice, John would likely have more pizza because Lee's slice has been reduced in size. However, without knowing the exact dimensions of the removed edge, we can't calculate the precise difference in area.

]

Answer:

First blank: 1/4

Second blank: 2/3

Step-by-step explanation:

[tex](x+\frac{1}{4})^2=\frac{4}{9}[/tex]

Applying the square root of both sides gives:

[tex](x+\frac{1}{4})=\pm \sqrt{\frac{4}{9}}[/tex]

[tex]x+\frac{1}{4}=\pm \frac{\sqrt{4}}{\sqrt{9}}[/tex]

[tex]x+\frac{1}{4}=\pm \frac{2}{3}[/tex]

The blanks are 1/4 and 2/3.

When we take the square root on both sides of the equation, then the whole square term becomes its square root, but the constant term on the other side has a ± sign as the square root of n can be -√n as well as √n, because the square of a negative number is also a positive number.

So the given equation is (x+1/4)² = 4/9

Taking square root on both sides we get

(x+1/4) = ±2/3 using the square root property of equality.

Hence the blanks are 1/4 and 2/3 of the given question.

Learn more about the property of equality here

https://brainly.com/question/11225212

#SPJ2

Answer:

Let's replace the h(x) function in g(x) and then use 74 as a result on the axis y. The correct answer is 4 .

Answer: y= -1/2x + 3/4

Step-by-step explanation: For a line in the for of y=mx + b, the slope is m and y intercept is b.

Answer:

y = - [tex]\frac{1}{2}[/tex] x + [tex]\frac{3}{4}[/tex]

Step-by-step explanation:

The equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

Rearrange 6x + 12y = 9 into this form

Subtract 6x from both sides

12y = - 6x + 9 ( divide all terms by 12 )

y = - [tex]\frac{6}{12}[/tex] x + [tex]\frac{9}{12}[/tex], that is

y = - [tex]\frac{1}{2}[/tex] x + [tex]\frac{3}{4}[/tex]

Answer:

[tex]-\frac34 -\frac14 i[/tex]

Step-by-step explanation:

Let's start with distributing: [tex]\frac1{4i}-\frac{3i}{4i}[/tex] Simplify i with i in the second term, and rearrange.[tex]-\frac34 +\frac1{4i}[/tex]. Since i in the denominator looks ugly, let's multiply top and bottom by i. [tex]-\frac34 +\frac i{4i^2} = -\frac34 +\frac i{4(-1)} =-\frac34 -\frac14 i[/tex]. The middle passage is based on the fact that, by definition, [tex]i^2=-1[/tex]

The total number of coins required to fill all the [tex]64[/tex] boxes are [tex]\boxed{\bf 18446744073709551615}[/tex].

Further explanation:

In a chessboard there are [tex]64[/tex] boxes.

The objective is to determine the total number of coins required to fill the [tex]64[/tex] boxes in chessboard.

In the question it is given that in the first box there is [tex]1[/tex] coin, in the second box there are [tex]2[/tex] coins, in the third box there are [tex]8[/tex] coins and it continues so on.

A sequence is formed for the number of coins in different boxes.

The sequence formed for the number of coins in different boxes is as follows:

[tex]\boxed{1,2,4,8,...}[/tex]

The above sequence can also be represented as shown below,

[tex]\boxed{2^{0},2^{1},2^{2},2^{3},...}[/tex]

It is observed that the above sequence is a geometric sequence.

A geometric sequence is a sequence in which the common ratio between each successive term and the previous term are equal.

The common ratio [tex](r)[/tex] for the sequence is calculated as follows:

[tex]\begin{aligned}r&=\dfrac{2^{1}}{2^{0}}\\&=2\end{aligned}[/tex]

The [tex]n^{th}[/tex] term of a geometric sequence is expressed as follows:

[tex]\boxed{a_{n}=ar^{n-1}}[/tex]

In the above equation [tex]a[/tex] is the first term of the sequence and [tex]r[/tex] is the common ratio.

The value of [tex]a[/tex] and [tex]r[/tex] is as follows:

[tex]\boxed{\begin{aligned}a&=1\\r&=2\end{aligned}}[/tex]

Since, the total number of boxes are [tex]64[/tex] so, the total number of terms in the sequence is [tex]64[/tex].

To obtain the number of coins which are required to fill the [tex]64[/tex] boxes we need to find the sum of sequence formed as above.

The sum of [tex]n[/tex] terms of a geometric sequence is calculated as follows:

[tex]\boxed{S_{n}=a\left(\dfrac{r^{n}-1}{r-1}\right)}[/tex]

To obtain the sum of the sequence substitute [tex]64[/tex] for [tex]n[/tex], [tex]1[/tex] for [tex]a[/tex] and [tex]2[/tex] for [tex]r[/tex] in the above equation.

[tex]\begin{aligned}S_{n}&=1\left(\dfrac{2^{64}-1}{2-1}\right)\\&=\dfrac{18446744073709551616-1}{1}\\&=18446744073709551615\end{aligned}[/tex]

Therefore, the total number of coins required to fill all the [tex]64[/tex] boxes are [tex]\boxed{\bf 18446744073709551615}[/tex].

Learn more:

1. A problem on greatest integer function https://brainly.com/question/8243712  

2. A problem to find radius and center of circle https://brainly.com/question/9510228  

3. A problem to determine intercepts of a line https://brainly.com/question/1332667  

Answer details:  

Grade: High school  

Subject: Mathematics  

Chapter: Sequence

Keywords: Series, sequence, logic, groups, next term, successive term, mathematics, critical thinking, numbers, addition, subtraction, pattern, rule., geometric sequence, common ratio, nth term.

Coins on the chessboard follow a doubling pattern. In the nth box, the coins can be expressed as [tex]\(2^{(n-1)}[/tex]. The total coins for all 64 boxes is [tex]2^{63}[/tex].

Certainly, let's break down the doubling pattern of coins in each chessboard box, expressed in exponential form:

1. **First Box (kotak pertama):

  - Number of coins: [tex]\(2^0 = 1\)[/tex] (2 raised to the power of 0).

2. **Second Box (kotak kedua):

  - Number of coins: [tex]\(2^1 = 2\)[/tex] (2 raised to the power of 1).

3. **Third Box (kotak ketiga):

  - Number of coins: [tex]\(2^2 = 4\)[/tex] (2 raised to the power of 2).

4. **Fourth Box (kotak keempat):

  - Number of coins: [tex]\(2^3 = 8\)[/tex] (2 raised to the power of 3).

The pattern continues, doubling the number of coins with each subsequent box.

For the n-th box, the number of coins is given by [tex]\(2^{(n-1)}[/tex], where n is the box number.

So, the exponential form for the number of coins in each chessboard box is [tex]\(2^{(n-1)}[/tex], where n is the box number ranging from 1 to 64.

For more such questions on chessboard:

https://brainly.com/question/29504883

#SPJ6

Que. A father challenges his child to calculate the total number of coins needed to fill a chessboard. In the first box, 1 coin is placed, 2 coins in the second box, 4 coins in the third, and so on, up to the 64th box. Help the child determine the doubling pattern of coins in each chessboard box, expressed in exponential form.

Final answer:

To calculate the volume of a cylinder with a radius of (x+4) cm and height of 5 cm, substitute the values into the formula V = πr²h, resulting in V = 5π(x² + 8x + 16), which gives the volume in standard form.

Explanation:

The question is about finding the volume of a cylinder with a given radius of (x+4) cm and a height of 5 cm. The formula to calculate the volume of a cylinder is V = πr²h, where 'V' is the volume, 'r' is the radius, and 'h' is the height of the cylinder.

To find the volume with the given dimensions, we substitute 'r' with (x + 4) and 'h' with 5. This results in:

V = π(x + 4)² × 5 = π(x² + 8x + 16) × 5

Simplifying this expression gives us:

V = 5π(x² + 8x + 16)

This is the volume of the cylinder in standard form, expressed as a function of x. Therefore, the volume depends on the value of x, and this expression allows us to calculate it for any given 'x'.

Answer:

Midpoint = (  (x1+x2)/2 , (y1+y2)/2 )

Step-by-step explanation:

Please upload the line segment otherwise, you can use the equation above to solve for it.

Answer:

  see below for the stretched graph; see the second attachment for the functions

Step-by-step explanation:

The transformation ...

  g(x) = f(x/a)

represents a horizontal stretch of f(x) by a factor of "a". You want a stretch by a factor of 4, so you can use a=4:

  g(x) = f(x/4)

_____

Horizontal stretch by a factor of 4 means all the points on the graph of g(x) are 4 times as far from they y-axis as they are on the graph of f(x). That is, x must be 4 times as large to give the same y-value.

The graphs of f(x) and g(x) are shown in the second attachment, along with their equations.

Answer:

a) ∀x∃y ¬∀zT(x, y, z)

∀x∃y ∃z ¬T(x, y, z)

b) ∀x¬[∃y (P(x, y) ∨ Q(x, y))]

∀x∀y ¬ [P(x, y) ∨ Q(x, y)]

∀x∀y [¬P(x, y) ^ ¬Q(x, y)]

c) ∀x ¬∃y (P(x, y) ^ ∃zR(x, y, z))

∀x ∀y ¬(P(x, y) ^ ∃zR(x, y, z))

∀x ∀y (¬P(x, y) v ¬∃zR(x, y, z))

∀x ∀y (¬P(x, y) v ∀z¬R(x, y, z))

d) ∀x¬∃y (P(x, y) → Q(x, y))

∀x∀y ¬(P(x, y) → Q(x, y))

∀x∀y (¬P(x, y) ^ Q(x, y))

Answer:

a) ∃x∀y∃z~T(x, y, z)

b) ∃x∀y~P(x, y) ∧ ∃x∀y~Q(x, y)

c) ∃x∀y(~P(x, y) ∨ ∀z~R(x, y, z))

d) ∃x∀y(P(x, y) → ~Q(x, y))

Step-by-step explanation:

The negation of a is written as ~a.

Note the following properties that are going to be applied in the problems here :

~(P → Q) = P → ~Q

De Morgan's Laws

~(P ∨ Q) = ~P ∧ ~Q

~(P ∧ Q) = ~P ∨ ~Q

~∃xP = ∀xP

~∀xP = ∃xP

So back to the original problem.

a) ∀x∃y∀zT(x, y, z)

We have the negation as

~[∀x∃y∀zT(x, y, z)]

= ∃x~∃y∀zT(x, y, z)

= ∃x∀y∀~zT(x, y, z)

= ∃x∀y∃z~T(x, y, z)

b) ∀x∃yP(x, y) ∨ ∀x∃yQ(x, y)

Negation is:

~[∀x∃yP(x, y) ∨ ∀x∃yQ(x, y)]

= ~∀x∃yP(x, y) ∧ ~∀x∃yQ(x, y)

= ∃x~∃yP(x, y) ∧ ∃x~∃yQ(x, y)

= ∃x∀y~P(x, y) ∧ ∃x∀y~Q(x, y)

c) ∀x∃y(P(x, y) ∧ ∃zR(x, y, z))

Negation is:

~[∀x∃y(P(x, y) ∧ ∃zR(x, y, z))]

= ~∀x∃y(P(x, y) ∧ ∃zR(x, y, z))

= ∃x~∃y(P(x, y) ∧ ∃zR(x, y, z))

= ∃x∀y~(P(x, y) ∧ ∃zR(x, y, z))

= ∃x∀y(~P(x, y) ∨ ~∃zR(x, y, z))

= ∃x∀y(~P(x, y) ∨ ∀z~R(x, y, z))

d) ∀x∃y(P(x, y) → Q(x, y))

Negation is:

~[∀x∃y(P(x, y) → Q(x, y))]

= ~∀x∃y(P(x, y) → Q(x, y))

= ∃x~∃y(P(x, y) → Q(x, y))

= ∃x∀y~(P(x, y) → Q(x, y))

= ∃x∀y(P(x, y) → ~Q(x, y))

Answer:

  b. f(x) = x³ +3x² -6x -18

Step-by-step explanation:

You want the cubic function with zeros ±√6 and -3.

Each zero p gives rise to a factor (x-p). This means the factored form of f(x) will be ...

  f(x) = (x -√6)(x +√6)(x -(-3))

Expanding this product gives ...

  f(x) = (x² -6)(x +3)

  f(x) = x³ +3x² -6x -18 . . . . . . . matches choice B

Answer:

  see below

Step-by-step explanation:

It's a little tough to draw on regular graph paper because the slope is so steep and the slope is not an integer. Shown below is the graph with a couple of the points labeled.

Answer:

31

Step-by-step explanation: