Please help find the linear pair

Step-by-step explanation:

We will prove by contradiction. Assume that [tex]2^n + 1[/tex] is an odd prime but n is not a power of 2. Then, there exists an odd prime number p such that [tex]p\mid n[/tex]. Then, for some integer [tex]k\geq 1[/tex],

[tex]n=p\times k.[/tex]

Therefore

Here we will use the formula for the sum of odd powers, which states that, for [tex]a,b\in \mathbb{R}[/tex] and an odd positive number [tex]n[/tex],

[tex]a^n+b^n=(a+b)(a^{n-1}-a^{n-2}b+a^{n-3}b^2-...+b^{n-1})[/tex]

Applying this formula in 1) we obtain that

[tex]2^n + 1=2^{p\times k} + 1=(2^{k})^p + 1^p=(2^k+1)(2^{k(p-1)}-2^{k(p-2)}+...-2^{k}+1)[/tex].

Then, as [tex]2^k+1>1[/tex] we have that [tex]2^n+1[/tex] is not a prime number, which is a contradiction.

In conclusion, if [tex]2^n+1[/tex] is an odd prime, then n must be a power of 2.

Answer:

There are 12 sets of four marbles include all the red ones.    

Step-by-step explanation:

Given : A bag contains three red marbles, two green ones, one lavender one, four yellows, and five orange marbles.

To find : How many sets of four marbles include all the red ones?

Solution :

Number of red marbles = 3

Number of green marbles = 2

Number of lavender marbles = 1

Number of yellow marbles = 4

Number of orange marbles = 5

We have to form sets of four marbles include all the red ones,

For position of getting red ones we have three red marbles i.e. [tex]^3C_3[/tex]

For the fourth one we have 12 choices i.e. [tex]^{12}C_1[/tex]

Total sets of four marbles include all the red ones is

[tex]=^3C_3\times ^{12}C_1[/tex]

[tex]=1\times 12[/tex]

[tex]=12[/tex]

Therefore, There are 12 sets of four marbles include all the red ones.

Answer:

D = 23.7 miles

Step-by-step explanation:

Given data:

[tex]\theta=84 degrees[/tex]

Time[tex] = 20 min = \frac{1}{3} hr[/tex]

Distance of A [tex]= (60 mi/hr)\times \frac{1}{3} hr =20 mi[/tex]

Distance of B =[tex] (45 mi/hr)\frac{1}{3} hr = 15 mi[/tex]

Draw a triangle.

By using cosine formula we can determine the distance between them

[tex]D^2 = 20^2 + 15^2 - 2\times 20\times 15\times cos(84)[/tex]

[tex]D^2 = 625 - 600 cos(84) [/tex]

[tex]D^2 = 625 - 63.2 [/tex]

[tex]D^2 = 561.8[/tex]

Thus D = 23.7 miles

Answer:  b. number of trees

Step-by-step explanation:

The concept of geometric probability is basically use when we have continuous data .

Since it is impossible to count continuous data , but geometrically ( in form of length, area etc) we can count the outcomes in general to calculate the required probability.

Therefore, from the given options , Option b. "number of trees" would not be used for geometric probability because among all it is the only discrete case which is countable.

Rest of items ( a. area of a rug , c. length of time , d. length of a field) would be used for geometric probability,

Answer: 10.6

Step-by-step explanation:

divide the number she spent (2.65) by the amount the store charges per ounce (.25)

2.65/ .25 = 10.6

Answer:

The cost of the old ball was $100.

Step-by-step explanation:

The cost of the new ball = $300

The new ball has three times the price of his old ball.

So, let the price of the old ball be = x

As per situation, we get the equation:

[tex]3x=300[/tex]

Dividing both sides by 3:

[tex]\frac{3x}{3}= \frac{300}{3}[/tex]

=> x = 100

Hence, the cost of the old ball was $100.

Answer:

See answer below

Step-by-step explanation:

In inductive reasoning we arrive to a general conclusion based on particular observations.

For example:

"My friends Peter and Mary study at XYZ University. Peter and Mary are brilliant.

Yesterday I met Joe in a party. Joe also studies at XYZ University and he happens to be brilliant, too.

I conclude that all the students of XYZ University are brilliant"

Notice that this kind of reasoning might lead to a false conclusion.

In deductive reasoning, we arrive to a particular conclusion based on general observations. So, deductive reasoning is the opposite of inductive reasoning.

An example of deductive reasoning could be the following:

"To be accepted in the XYZ University you must pass a test with a score greater than 70%.

My friend Peter studies at XYZ University, so he passed the test with a score greater than 70%"

In deductive reasoning you may arrive to a false conclusion if your general assumption is false.

Answer:

but did now am the am

Step-by-step explanation:

Answer:

[tex]A=128[/tex]

Step-by-step explanation:

First of all we need to graph f(x)=8x, (First picture)

Now we have to calculate the area enclosed by the graph of the function, the horizontal axis, and vertical lines at [tex]x_{1}=2[/tex] and [tex]x_{2}=6[/tex] ,

The area that we have to calculate is in pink (second picture).

The function is positive and the domain is [tex][2,6][/tex] then we can calculate the area with this formula:

[tex]A=\int\limits^b_a {f(x)} \, dx[/tex],

In this case [tex]b=x_{2} , a=x_{1}[/tex]

[tex]A=\int\limits^6_2 {8x} \, dx = 8\int\limits^6_2 {x} \, dx[/tex]

The result of the integral is,

[tex]A=8\frac{x^{2}}{2}[/tex], but the integral is defined in [2,6] so we have to apply Barrow's rule,

Barrow's rule:

If f is continuous in [a,b] and F is a primitive of f in [a,b], then:

[tex]\int\limits^b_a {f(x)} \, dx =F(b)-F(a)[/tex]

Applying Barrow's rule the result is:

[tex]A=8.\frac{6^{2} }{2}-8.\frac{2^{2} }{2}[/tex]

[tex]A=8.\frac{36}{2} -8.\frac{4}{2}[/tex]

[tex]A=144-16[/tex]

[tex]A=128[/tex]

Answer: Each person will take [tex]\dfrac{1}{6}[/tex] of the pie.

Step-by-step explanation:

Given : One pie is shared equally by 6 people.

The total number of persons = 6

Now, the fraction of the whole pie each person will take :-

[tex]\dfrac{\text{Number of pie}}{\text{total persons}}\\\\=\dfrac{1}{6}[/tex]

Therefore, the fraction of the whole pie each person will take= [tex]\dfrac{1}{6}[/tex]

Answer:

B. [tex]1.83\text{ m}^2[/tex]

Step-by-step explanation:

We are asked to find the body surface area of a person whose height is 150 cm and who weighs 80 kg.

[tex]\text{Body surface area}( m^2)=\sqrt{\frac{\text{Height (cm)}\times \text{Weight (kg)}}{3600}}[/tex]

Substitute the values:

[tex]\text{Body surface area}( m^2)=\sqrt{\frac{150\text{ cm}\times 80\text{(kg)}}{3600}}[/tex]

[tex]\text{Body surface area}( m^2)=\sqrt{\frac{12,000}{3600}}[/tex]

[tex]\text{Body surface area}( m^2)=\sqrt{3.3333333}[/tex]

[tex]\text{Body surface area}( m^2)=1.825741[/tex]

[tex]\text{Body surface area}( m^2)=1.83[/tex]

Therefore, the body surface area of the person would be 1.83 square meters.

Answer: $ 67.03

Step-by-step explanation:

Given : The average expenditure in a sample survey of 50 male consumers was $135.67, and the average expenditure in a sample survey of 38 female consumers was $68.64.

i.e. [tex]\overline{x}_1=\$135.67\ \ \&\ \ \overline{x}_2=\$68.64[/tex]

The best point estimate of the difference between the two population means is given by :-

[tex]\overline{x}_1-\overline{x}_2\\\\=135.67-68.64=67.03[/tex]

Hence, the point estimate of the difference between the population mean expenditure for males and the population mean expenditure for females : $ 67.03

The point estimate of the difference between the average expenditure of male and female consumers for Valentine's Day is $67.03.

The subject of your question is related to comparative statistical analysis between two groups, in this case, male and female consumers on Valentine's Day expenditures. Your question focuses on finding the point estimate for the difference between the population mean expenditure of males and females.

The point estimate is calculated by simply subtracting one mean from the other. According to your data, the average expenditure of the male consumers is $135.67 and of female consumers is $68.64. So, the calculation looks like this: $135.67 - $68.64 = $67.03. Therefore, the point estimate of the difference between the population mean expenditure for males and the population mean expenditure for females is $67.03.

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Answer:

B. the 1101 people interviewed.

Step-by-step explanation:

An opinion poll contacts 1101 adults and asks them, " Which political party do you think has better ideas for leading the country in the 21st century?"

The sample in this setting is the 1101 people interviewed.

Here sample space is 1101.

Answer:

B. the 1101 people interviewed.

Step-by-step explanation:

The sample is the subset of the population and it takes for making the experiment easy. Further, the sample is said to be best if it is the representation of the whole population.

Here, since opinion poll contacts all 1101 adults for knowing their opinion about political parties.

Hence the sample space is all 1101 people interviewed.

Rewriting the expression using m=2p we have:

Answer:

[tex]m^{2} +5 -1[/tex] is an odd integer but the converse is not true.

Step-by-step explanation:

Even numbers are written as 2n where n is any integer, while odd numbers are written as 2n-1 where n is any integer.

a) To prove that [tex]m^{2} +5m-1[/tex] is an odd integer, we have to prove that it can be written as 2n-1.

By hypothesis, m is an even integer so we will write it as 2p.

Rewriting the original expression using [tex]m=2p[/tex] we have:

[tex]m^{2} +5m-1 = (2p)^{2} +5(2p)-1[/tex]

Solving the expression and factorizing it we get

[tex]4p^{2} +10p -1 = 2(2p^{2}+5p) -1\\ \\[/tex]

And this last expression is an expression of the form 2n-1, and therefore [tex]m^{2} +5m-1[/tex] is an odd integer.

b) The converse would be: if [tex]m^{2} +5m-1[/tex] is an odd integer, then m is an even integer.

We'll give a counterexample, let's make [tex]m=3[/tex], then

[tex]m^{2} +5m-1[/tex]

[tex]3^{2}+5(3)-1 = 23[/tex] is an odd integer but m is odd.

Therefore, the converse is not true.

Answer:

Step-by-step explanation:

Given that the first difference of a sequence is the arithmetic sequence 1​, 3​, 5​, 7​, 9​, ....

a) When I term a =1

[tex]a_2 =1+1 =2\\a_3 = 4+5 =9\\a_4 = 9+7 =16\\a_5 =16+9 =25\\a_6=25+11 =36[/tex]

Thus first 6 terms are

1,2,5,12,21,32.....

b) Here [tex]a_1+a_2=5\\a_2-a_1 =3\\-------------\\2a_2=8\\a_2 =4\\a_1 =1[/tex]

[tex]a_2 =1+3 =4\\a_3 = 4+5 =9\\a_4 = 9+7 =16\\a_5 =16+9 =25\\a_6=25+11 =36[/tex]

So sequence would be

3,4,9,16,25, 36,...

c) When 5th term is 28

we have the sequences as

a1, a1+1,a1+1+3, ...a1+1+3+5+7

When 5th term is 28 we have

[tex]a_1 +16 =28\\a_1 =12\\[/tex]

Hence first 6 terms would be

12, 13, 16, 21, 28, 37,...

Answer:   [tex]\dfrac{511}{512}[/tex]

Step-by-step explanation:

We know that the total number of faces on a coin [Tail, Heads] =2

The probability of getting a tail = [tex]\dfrac{1}{2}=\dfrac{1}{2}[/tex]

The  probability of getting no tail = [tex]1-\dfrac{1}{2}=\dfrac{1}{2}[/tex]

The "at least once" rule says that the when a coin is tossed n times , then the probability of getting at least one tail is given by :-

[tex]\text{P(Atleast one tail)}=1-(\text{P(No tail)})^n[/tex]

Since , n=9

Then, the probability of getting at least one tail is given by :-

[tex]\text{P(Atleast one tail)}=1-(\text{P(No tail)})^9\\\\=1-(\dfrac{1}{2})^9\\\\=1-\dfrac{1}{512}\\\\=\dfrac{511}{512}[/tex]

The probability is [tex]\dfrac{511}{512}[/tex]

The probability of getting at least one tail when tossing nine fair coins is 511/512. This is computed by finding the complement of getting no tails (only heads) when flipping the coins.

In this problem, we are looking for the probability of getting at least once a tail when tossing nine fair coins. To solve it, we use the rule of complementary probability. Rather than finding the probability for one or more tails, it's easier to find the probability of the complement, which is getting no tails (only heads). With a fair coin, the probability of getting a head on one toss is 1/2, so for nine tosses, it's (1/2)^9 = 1/512. However, this is the probability of getting no tails at all. We want the exact opposite; at least one tail. So, we subtract this probability from 1: P(at least one tail) = 1 - P(no tails) = 1 - 1/512 = 511/512.

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Answer:

D. n is divisible by 6 and n is NOT divisible by both 2 and 3.

Step-by-step explanation:

The statement we want to negate is:

"IF n is divisible by 6 THEN n is divisible by both 2 and 3"

you should think of it as having one single antecedent (the sentence that follows after the IF) and one single consequent (the sentence that follows after THEN), as:

"IF ( n is divisible by 6 )THEN ( n is divisible by both 2 and 3 )"

This kind of statements are negated by saying that the antecedent is true, but the consequent isn't true (which is kind of saying that the antecedent being true doesn't necessarily make the consequent also true).

So the negation of the original statement is just:

"(n is divisible by 6) and NOT (n is divisible by both 2 and 3)"

which in common english is just

"n is divisible by 6 and n is NOT divisible by both 2 and 3."

Answer:

hippioo

Step-by-step explanation:

hippi hipppokmkmkouse

Answer:

102.8571 :)

Step-by-step explanation:

Answer:102.857 or rounded to 103

Step-by-step explanation:

You divide 720 by 7 which = 102.857

If it asks for a rounded number it would be 103

Answer with explanation:

For any object having the velocity vector as

[tex]\overrightarrow{v}=v_x\widehat{i}+v_y\widehat{j}+v_z\widehat{k}[/tex]

the magnitude of velocity is given by

[tex]|v|=\sqrt{v_x^2+v_y^2+v_z^2}[/tex]

For car 1 the velocity vector is

[tex]\overrightarrow{v}_1=20\widehat{i}+25\widehat{j}[/tex]

Therefore

[tex]|v_1|=\sqrt{20^2+25^2}\\\\\therefore v_1=32.0156units[/tex]

Similarly for car 2 we have

[tex]\overrightarrow{v}_2=30\widehat{i}[/tex]

Therefore

[tex]|v_2|=\sqrt{30^2}\\\\\therefore v_2=30.0units[/tex]

Comparing both the values we find that car 1 has the greater speed.

The correct answer is that the car with velocity vector 30i is traveling faster, and its speed is 30 units.

To determine which car is traveling faster, we need to calculate the magnitude (or speed) of each velocity vector. The magnitude of a velocity vector in two dimensions is given by the square root of the sum of the squares of its components.

For the first car, the velocity vector is[tex]\( \mathbf{v}_1 = 20\mathbf{i} + 25\mathbf{j} \). The magnitude of this vector is calculated as follows:\[ ||\mathbf{v}_1|| = \sqrt{(20)^2 + (25)^2} = \sqrt{400 + 625} = \sqrt{1025} \][/tex]

For the second car, the velocity vector is [tex]\( \mathbf{v}_2 = 30\mathbf{i} + 0\mathbf{j} \).[/tex]

The magnitude of this vector is calculated as:[tex]\[ ||\mathbf{v}_2|| = \sqrt{(30)^2 + (0)^2} = \sqrt{900 + 0} = \sqrt{900} = 30 \]Now, comparing the magnitudes of the two vectors:\[ ||\mathbf{v}_1|| = \sqrt{1025} \approx 32.0156 \]\[ ||\mathbf{v}_2|| = 30 \]It is clear that \( ||\mathbf{v}_1|| \) is approximately 32.0156 units, while \( ||\mathbf{v}_2|| \)[/tex] is exactly 30 units. Since 32.0156 is greater than 30, the car with velocity vector[tex]\( \mathbf{v}_2 = 30\mathbf{i} \)[/tex]  is traveling faster.Therefore, the faster car is traveling at a speed of 30 units.

Answer:

There are 47 corrupt senators and 2 honest senators.

Step-by-step explanation:

Freedonia has 49 senators. Each senator is either honest or corrupt.

At least 2 of the Freedonian senators are honest and that, out of any 3 Freedonian senators, at least 1 is corrupt.

As we can see that there are no 3 senators where all of them are honest. So either there is one senator who is corrupt and 2 are honest.

Also it is given that at least two of the Freedonian senators are honest.

Hence, we can conclude there are 2 honest senators and 47 corrupt senators.

Answer:

1200 ml per hour

Step-by-step explanation:

To compute what rate is 300 ml over 15 mins as a rate of ml per hour, we do a rule of 3, using a variable x as that amount of ml we don't know yet. We should have everything in the same units, so instead of writing 1 hour we write 60 minutes:

[tex] \frac{300~ml}{15~min}=\frac{x~ml}{60~mins}[/tex]

Now we solve for x:

[tex] \frac{300~ml}{15~min}\cdot 60~mins=x~ml[/tex]

[tex] \frac{18000~ml}{15} =x~ml[/tex]

[tex] 1200~ml =x~ml[/tex]

And so, now that we know the value of x, the rate we wanted to find is

[tex]\frac{1200~ml}{60~mins}[/tex]

Which is just 1200 ml per hour.

Answer:

(a), (a,c), (a,d), (a,e), (a,c,d), (a,c,e), (a,d,e), (a,c,d,e)

Step-by-step explanation:

We are given the set (a,b,c,d,e).

Total number of subsets of the above set are [tex]2^5[/tex] = 32

Subsets:

φ

(a,b,c,d,e)

(a), (b), (c), (d), (e)

(a,b), (a,c), (a,d), (a,e), (b,c), (b,d), (b,e), (c,d), (c,e), (d,e)

(a,b,c), (a,b,d), (a,b,e), (a,c,d), (a,c,e), (a,d,e), ( b,c,d), (b,c,e), (b,d,e), (c,d,e)

(a,b,c,d), (a,b,c,e), (a,b,d,e), (a,c,d,e), (b,c,d,e)

Subset having a but not b :

(a), (a,c), (a,d), (a,e), (a,c,d), (a,c,e), (a,d,e), (a,c,d,e)

Answer:

C. 36 in.

Step-by-step explanation:

The wall is 66 inches high by 72 inches wide.

The tiles are

a) 9 inch square = 3 by 3

b) 16 inch square = 4 by 4

c) 36 inch square = 6 by 6

d) 64 inch square = 8 by 8

Factors of 66 = 2 x 3 x 11

Factors of 72 = 2 x 2 x 2 x 3 x 3

Now, we can see that in both 66 and 72 , we have 2 x 3 common that is 6.

And square of 6 is 36.

So, the answer is option C.

Answer:

The kinetic energy of car with mass 1500 kg and with speed of 35 miles/hour is KE=183598 J and when the car increases its speed to 70 miles/hour the kinetic energy changes by a factor of 4.

Step-by-step explanation:

The first step is to convert the speed miles/hour to m/s.

[tex]35\frac{miles}{hour} *\frac{1609.34 \>m}{1 \>miles}*\frac{1 \>hour}{3600 \> s}=15.646 \frac{m}{s}[/tex]

Next, the formula for the kinetic energy is

[tex]KE=\frac{1}{2} mv^{2}[/tex]

So input the values given:

[tex]KE=\frac{1}{2} (1500)(15.646)^{2}\\KE=750 \cdot (15.646)^{2}\\KE=183597.987 = 183598 \frac{kg \cdot \>m^{2}}{s^{2}} \\KE=183598 \>J[/tex]

Notice that the speed of 70 miles/hour is the double of 35 miles/hour so we can say that [tex]v_{2}=2v_{1}[/tex] and use the formula for the kinetic energy

[tex]KE_{2} =\frac{1}{2} m(v_{2}) ^{2}\\if \: v_{2}= 2v_{1}, then \:\\KE_{2} =\frac{1}{2} m(2v_{1}) ^{2}\\KE_{2} =\frac{1}{2} m4(v_{1})^{2}\\KE_{2} =4(\frac{1}{2} m(v_{1})^{2})\\We \:know \:that \:KE_{1} =\frac{1}{2} m(v_{1})^{2} so\\KE_{2} =4(KE_{1})[/tex]

We can see that when the car increases its speed to 70 miles/hour the kinetic energy changes by a factor of 4.

Answer:

1) Prime Factorization

2) Technology

3) Existence of prime number in nature

Step-by-step explanation:

Prime numbers are the numbers whose divisors are 1 and the number itself, For example: 2, 3, 7, 11,...

Prime Numbers are a significant part of our life and are widely used in daily life.

1) Prime Factorization

This method help us to break a number into products of prime Number. This approach help us to find the LCM(Lowest Common Multiple) and GCD(Greatest Common Divisor)

2) Technology

Prime factorization forms the basis oh cryptography. Prime numbers play an important role in password protection and security purposes. They give the basis for many cryptographic algorithms.

3) Existence of prime number in nature

Many scientist have claimed that prime numbers exist in our life in unexpected form. For example, the number of petals in a flower, number of hexes in beehive, the pattern in pineapple are all related to prime number.

Answer:

12 ; 12 dollars

Step-by-step explanation:

Data provided in the question:

Revenue function, R = 12x

R is in dollars

Now,

The slope can be found out by differentiating the above revenue function w.r.t 'x'

thus,

[tex]\frac{\textup{dR}}{\textup{dx}}[/tex]= [tex]\frac{\textup{d(12x)}}{\textup{dx}}[/tex]

or

slope = 12

Now, for the second case of selling one more unit i.e x = 1, the revenue can be obtained by substituting x = 1 in revenue function

therefore,

R = 12 × 1 = 12 dollars

Answer:  Radius = 10 cm and Arc length = 5 cm

Step-by-step explanation:

The area of a sector with radius r and central angle [tex]\theta[/tex] (In radian) is given by :-

[tex]A=\dfrac{1}{2}r^2\theta[/tex]

Given : A sector of a circle has area [tex]25 cm^2[/tex] and central angle  0.5 radians.

Let r be the radius , then we have

[tex]25=\dfrac{1}{2}r^2(0.5)\\\\\Rightarrow\ r^2=\dfrac{2\times25}{0.5}\\\\\Rightarrow\ r^2=\dfrac{50}{0.5}=100\\\\\Rightarrow\ r=\sqrt{100}=10\ cm[/tex]

Thus, radius = 10 cm

The length of arc is given by :-

[tex]l=r\theta=10\times0.5=5\ cm[/tex]

Hence, the length of the arc = 5 cm

Final answer:

To find the radius and arc length of a sector of a circle, we use formulas related to the circumference and central angle of a circle. The radius of the sector is 10 cm and the arc length is 5 cm.

Explanation:

To find the radius and arc length of a sector of a circle, we need to use the formulas related to the circumference and central angle of a circle. The formula for the area of a sector is given by:

Area = (θ/2) * r^2

where θ is the central angle and r is the radius of the circle. We are given that the area of the sector is 25 cm^2 and the central angle is 0.5 radians. Setting up this equation, we get:

25 = (0.5/2) * r^2

Simplifying, we find:

r^2 = 100

Taking the square root of both sides, we find:

r = 10 cm

To find the arc length, we use the formula:

Arc Length = θ * r

Substituting the values, we find:

Arc Length = 0.5 * 10 = 5 cm

Answer:

The correct option is 5) {Paul, Ringo, Pete, John, George, Stuart}.

Step-by-step explanation:

Consider the provided sets:

B = {John, Paul, George, Ringo, Pete, Stuart}

Two sets are equal if all the elements of Sets are same.

Set B has the elements: John, Paul, George, Ringo, Pete and Stuart

Now consider the provided options of sets.

From the provided options of set only option 4 has all the elements of set B but the order is different.

Thus, the correct option is 5) {Paul, Ringo, Pete, John, George, Stuart}.

Answer: 6.416%

Step-by-step explanation:

The percentage error formula is given by :-

[tex]\%\text{error}=\dfrac{|\text{Estimate-Actual}|}{\text{Actual}}\times100[/tex]

Given : The estimated weight of a substance = 475 mg

The actual weight of the substance = 445 mg

Then,

[tex]\%\text{error}=\dfrac{| 475-445|}{445}\times100\\\\=\dfrac{30}{445}\times100=6.74157303371\approx6.416\%[/tex]

Hence, the percentage error in the first weighing. = 6.416%

Final answer:

The percentage error in the pharmacist's first weighing is approximately 6.74%, calculated by subtracting the accurate weight from the inaccurate weight, resulting in an absolute error of 30 mg, and then dividing the absolute error by the accurate weight, multiplying by 100.

Explanation:

To calculate the percentage error of the pharmacist's initial weighing, we first need to determine the absolute error by subtracting the accurate weight from the inaccurate weight. In this instance, the initial weight ( A ) recorded was 475 mg, and upon checking with a high accuracy balance, the true weight ( B ) was found to be 445 mg. Therefore, the absolute error ( Δ ) is the difference between these two measurements:  Δ =  A -  B = 475 mg - 445 mg = 30 mg.

After determining the absolute error, we can calculate the percentage error using the following formula:

Percentage Error = ( Δ /  B ) × 100%

Substituting in the respective values, we get:

Percentage Error = (30 mg / 445 mg) × 100% ≈ 6.74%

Answer:

The system has solution when:

[tex]a\neq 16[/tex]

The system has no solution when:

[tex]a=16[/tex]

Step-by-step explanation:

First rewrite the system in its augmented matrix form

[tex]\left[\begin{array}{cccc}1&2&-3&4\\3&-1&5&2\\4&1&a-14&a+2\end{array}\right][/tex]

Let´s apply row reduction process to  its associated augmented matrix:

[tex]F2-3F1\\F3-4F1[/tex]

[tex]\left[\begin{array}{cccc}1&2&-3&4\\0&-7&14&-10\\0&-7&a-2&a-14\end{array}\right][/tex]

[tex]F3-F2[/tex]

[tex]\left[\begin{array}{cccc}1&2&-3&4\\0&-7&14&-10\\0&0&a-16&a-4\end{array}\right][/tex]

Now we have this:

[tex]x+2y-3z=4\\0-7y+14=-10\\0+0+(a-16)z=a-4[/tex]

We can conclude now:

The system has no solution when:

[tex]a=16[/tex]

And the system has solution when:

[tex]a\neq 16[/tex]