What type of triangle has side lengths 9, 10, and √130? A. obtuse B. not a triangle C. acute D. right

Answer:

3x+2+x-3+2x+1+2(2x+5)=360

10x+10=360

x=35

The probability that the 4 oldest people in the group were selected is based on combinatorics is 0.00303 or 0.303%.

Given that:

Find how many ways the 4 oldest people can be selected from the group.

Since the 4 oldest people are already determined, there is only 1 way to select them.

To calculate the probability, to determine the total number of ways to select 4 people from the group of 11. This can be found using the combination formula:

Number of ways to choose k items from n items :

C(n,k) = n! / (k!(n-k)!)

Calculate the total number of ways to select 4 people from the group:

Plugging n and k value from given data:

C(11,4 )= 11! / (4!(11-4)!)

On simplifications gives:

C(11, 4) = 330.

Calculate the probability:

Probability = Number of ways 4 oldest people selected / Total number of ways to select 4 people

Plugging the given data:

Probability = 1 / 330

Probability ≈ 0.00303 or 0.303%.

Therefore, the  probability that the 4 oldest people in the group were selected is based on combinatorics is 0.00303 or 0.303%.

Learn more about probabilities here:

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Answer:

The value of the expression when [tex]m = 3.48[/tex] and [tex]b = 96.52[/tex] is 323.779.

Step-by-step explanation:

Let be [tex]f(m, b) = m\cdot b - m^{2}[/tex], if [tex]m = 3.48[/tex] and [tex]b = 96.52[/tex], the value of the expression:

[tex]f(3.48,96.52) = (3.48)\cdot (96.52)-3.48^{2}[/tex]

[tex]f(3.48,96.52) = 323.779[/tex]

The value of the expression when [tex]m = 3.48[/tex] and [tex]b = 96.52[/tex] is 323.779.

Answer:

The removed number is 11.

Step-by-step explanation:

Given that the average of 5 numbers is 7. So you have to find the total values of 5 numbers :

[tex]let \: x = total \: values[/tex]

[tex] \frac{x}{5} = 7[/tex]

[tex]x = 7 \times 5[/tex]

[tex]x = 35[/tex]

Assuming that the total values of 5 numbers is 35. Next, we have to find the removed number :

[tex]let \: y = removed \: number[/tex]

[tex] \frac{35 - y}{4} = 6[/tex]

[tex]35 - y = 6 \times 4[/tex]

[tex]35 - y = 24[/tex]

[tex]35 - 24 = y[/tex]

[tex]y = 11[/tex]

Okay, let's slightly generalize this

Average of [tex]n[/tex] numbers is [tex]a[/tex]

and then [tex]r[/tex] numbers are removed, and you're asked to find the sum of these [tex]r[/tex] numbers.

Solution:

If average of [tex]n[/tex] numbers is [tex]a[/tex] then the sum of all these numbers is [tex]n\cdot a[/tex]

Now we remove [tex]r[/tex] numbers, so we're left with [tex](n-r)[/tex] numbers. and their. average will be [tex]{\text{sum of these } (n-r) \text{ numbers} \over (n-r)}[/tex] let's call this new average [tex] a^{\prime}[/tex]

For simplicity, say, sum of these [tex]r[/tex] numbers, which are removed is denoted by [tex]x[/tex] .

so the new average is [tex]\frac{\text{Sum of } n \text{ numbers} - x}{n-r}=a^{\prime}[/tex]

or, [tex] \frac{n\cdot a -x}{n-r}=a^{\prime}[/tex]

Simplify the equation, and solve for [tex]x[/tex] to get,

[tex] x= n\cdot a -a^{\prime}(n-r)=n(a-a^{\prime})+ra^{\prime}[/tex]

Hope you understand it :)

Answer:

There is enough evidence to reject the highway department executive's claim about the distribution of fatal accidents between each month, as the Variance is 14 and the Standard Deviation = 4 approximately.

There is a high degree of variability in the mean of the population as explained by the Variance and the Standard Deviation.

Step-by-step explanation:

Month       No. of              Mean       Squared

           Fatal Accidents  Deviation   Difference

Jan          8                       -4                  16

Feb        15                        3                   9

Mar         9                       -3                   9

Apr         8                       -4                  16

May       13                        1                    1

Jun         6                      -6                 36

Jul         17                       5                 25

Aug       15                       3                   9

Sep       10                      -2                   4

Oct        9                       -3                   9

Nov    18                          6                 36

Dec    12                          0                   0

Total 140                                         170

Mean = 140/12 = 12    Mean of squared deviation (Variance) = 170/12 = 14.16667

Standard deviation = square root of variance = 3.76386 = 4

The fatal accidents' Variance is a measure of how spread out the fatal accident data set is. It is calculated as the average squared deviation of the number of each month's accident from the mean of the fatal accident data set.  It also shows how variable the data varies from the mean of approximately 12.

The fatal accidents' Standard Deviation is the square root of the variance, and a useful measure of variability when the distribution is normal or approximately normal.

Answer:

$2,589.52

Step-by-step explanation:

[tex] A = P(1 + \dfrac{r}{n})^{nt} [/tex]

We start with the compound interest formula above, where

A = future value

P = principal amount invested

r = annual rate of interest written as a decimal

n = number of times interest is compound per year

t = number of years

For this problem, we have

P = 2000

r = 0.026

n = 2

t = 10,

and we find A.

[tex] A = $2000(1 + \dfrac{0.026}{2})^{2 \times 10} [/tex]

[tex] A = $2589.52 [/tex]

Compound interest formula:

Total = principal x ( 1 + interest rate/compound) ^ (compounds x years)

Total = 2000 x 1+ 0.026/2^20

Total = $2,589.52

Answer:

C) A only

Step-by-step explanation:

In statistics, the null hypothesis is the default hypothesis and the alternative hypothesis is  the research hypothesis. The alternative hypothesis usually comes in place to challenge the null hypothesis in order to determine if the test is statistically significant or not.

Similarly,

In hypothesis testing, the confidence interval consist of all reasonable value of the population mean. Values for which the null hypothesis will be rejected [tex]H_o[/tex] .

Given that:

At 95% confidence interval for the  difference between the population means is (1.4, 8.7).

The level of significance = 1 - 0.95 = 0.05  = 5%

So , If the hypothesis test is based on the same samples, The hypothesis µ1 = µ2 would be rejected at the 5% level of significance.

Answer:

1078

Step-by-step explanation:

The product of 22 and 49 is 1078.

Answer:

1078 is the product

Step-by-step explanation:

Answer:

Step-by-step explanation:

Given that:

[tex]T(x,y) = \dfrac{100}{1+x^2+y^2}[/tex]

This implies that the level curves of a function(f) of two variables relates with the curves with equation f(x,y) = c

here c is the constant.

[tex]c = \dfrac{100}{1+x^2+2y^2} \ \ \--- (1)[/tex]

By cross multiply

[tex]c({1+x^2+2y^2}) = 100[/tex]

[tex]1+x^2+2y^2 = \dfrac{100}{c}[/tex]

[tex]x^2+2y^2 = \dfrac{100}{c} - 1 \ \ -- (2)[/tex]

From (2); let assume that the values of c > 0 likewise c < 100, then the interval can be expressed as 0 < c <100.

Now,

[tex]\dfrac{(x)^2}{\dfrac{100}{c}-1 } + \dfrac{(y)^2}{\dfrac{50}{c}-\dfrac{1}{2} }=1[/tex]

This is the equation for the  family of the eclipses centred at (0,0) is :

[tex]\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1[/tex]

[tex]a^2 = \dfrac{100}{c} -1 \ \ and \ \ b^2 = \dfrac{50}{c}- \dfrac{1}{2}[/tex]

Therefore; the level of the curves are all the eclipses with the major axis:

[tex]a = \sqrt{\dfrac{100 }{c}-1}[/tex]  and a minor axis [tex]b = \sqrt{\dfrac{50 }{c}-\dfrac{1}{2}}[/tex]  which satisfies the values for which 0< c < 100.

The sketch of the level curves can be see in the attached image below.

Answer:

[tex]\huge \boxed{\mathrm{\$ \ 7,533.33}}[/tex]

Step-by-step explanation:

There are 12 months in one whole year.

In one year, the person earns $96,600 with bonus.

The person gets a bonus of $6,200 during Christmas.

96,600 - 6,200 = 90,400

The person earns $90,400 yearly.

[tex]\frac{90,400}{12}[/tex] = 7,533.3333

Each month, the person earns $7,533.33, to the nearest cent.

Answer:

Solution : 5.244 - 16.140i

Step-by-step explanation:

If we want to express the two as a product, we would have the following expression.

[tex]-6\left[\cos \left(\frac{2\pi }{5}\right)+i\sin \left(\frac{2\pi }{5}\right)\right]\cdot 2\sqrt{2}\left[\cos \left(\frac{-\pi }{2}\right)+i\sin \left(\frac{-\pi \:}{2}\right)\right][/tex]

Now we have two trivial identities that we can apply here,

( 1 ) cos(- π / 2) = 0,

( 2 ) sin(- π / 2) = - 1

Substituting them,

= [tex]-6\cdot \:2\sqrt{2}\left(0-i\right)\left(\cos \left(\frac{2\pi }{5}\right)+i\sin \left(\frac{2\pi }{5}\right)\right)[/tex]

= [tex]-12\sqrt{2}\sin \left(\frac{2\pi }{5}\right)+12\sqrt{2}\cos \left(\frac{2\pi }{5}\right)i[/tex]

Again we have another two identities we can apply,

( 1 ) sin(x) = cos(π / 2 - x )

( 2 ) cos(x) = sin(π / 2 - x )

[tex]\sin \left(\frac{2\pi }{5}\right)=\cos \left(\frac{\pi }{2}-\frac{2\pi }{5}\right) = \frac{\sqrt{2}\sqrt{5+\sqrt{5}}}{4}[/tex]

[tex]\cos \left(\frac{2\pi }{5}\right)=\sin \left(\frac{\pi }{2}-\frac{2\pi }{5}\right) = \frac{\sqrt{2}\sqrt{3-\sqrt{5}}}{4}[/tex]

Substitute,

[tex]-12\sqrt{2}(\frac{\sqrt{2}\sqrt{5+\sqrt{5}}}{4}) + 12\sqrt{2}(\frac{\sqrt{2}\sqrt{3-\sqrt{5}}}{4})[/tex]

= [tex]-6\sqrt{5+\sqrt{5}}+6\sqrt{3-\sqrt{5}} i[/tex]

= [tex]-16.13996 + 5.24419i[/tex]

= [tex]5.24419i - 16.13996[/tex]

As you can see option d is the correct answer. 5.24419 is rounded to 5.244, and 16.13996 is rounded to 16.14.

Answer:

red

Step-by-step explanation:

Since the bag contains more red marbles than any other color, you are most likely to pick a red marble

Answer and Step-by-step explanation: Spherical coordinate describes a location of a point in space: one distance (ρ) and two angles (Ф,θ).To transform cartesian coordinates into spherical coordinates:

[tex]\rho = \sqrt{x^{2}+y^{2}+z^{2}}[/tex]

[tex]\phi = cos^{-1}\frac{z}{\rho}[/tex]

For angle θ:

Calculating:

a) (4,2,-4)

[tex]\rho = \sqrt{4^{2}+2^{2}+(-4)^{2}}[/tex] = 6

[tex]\phi = cos^{-1}(\frac{-4}{6})[/tex]

[tex]\phi = cos^{-1}(\frac{-2}{3})[/tex]

For θ, choose 1st option:

[tex]\theta = tan^{-1}(\frac{2}{4})[/tex]

[tex]\theta = tan^{-1}(\frac{1}{2})[/tex]

b) (0,8,15)

[tex]\rho = \sqrt{0^{2}+8^{2}+(15)^{2}}[/tex] = 17

[tex]\phi = cos^{-1}(\frac{15}{17})[/tex]

[tex]\theta = tan^{-1}\frac{y}{x}[/tex]

The angle θ gives a tangent that doesn't exist. Analysing table of sine, cosine and tangent: θ = [tex]\frac{\pi}{2}[/tex]

c) (√2,1,1)

[tex]\rho = \sqrt{(\sqrt{2} )^{2}+1^{2}+1^{2}}[/tex] = 2

[tex]\phi = cos^{-1}(\frac{1}{2})[/tex]

[tex]\phi[/tex] = [tex]\frac{\pi}{3}[/tex]

[tex]\theta = tan^{-1}\frac{1}{\sqrt{2} }[/tex]

d) (−2√3,−2,3)

[tex]\rho = \sqrt{(-2\sqrt{3} )^{2}+(-2)^{2}+3^{2}}[/tex] = 5

[tex]\phi = cos^{-1}(\frac{3}{5})[/tex]

Since x < 0, use 2nd option:

[tex]\theta = \pi + tan^{-1}\frac{1}{\sqrt{3} }[/tex]

[tex]\theta = \pi + \frac{\pi}{6}[/tex]

[tex]\theta = \frac{7\pi}{6}[/tex]

Cilindrical coordinate describes a 3 dimension space: 2 distances (r and z) and 1 angle (θ). To express cartesian coordinates into cilindrical:

[tex]r=\sqrt{x^{2}+y^{2}}[/tex]

Angle θ is the same as spherical coordinate;

z = z

Calculating:

a) (4,2,-4)

[tex]r=\sqrt{4^{2}+2^{2}}[/tex] = [tex]\sqrt{20}[/tex]

[tex]\theta = tan^{-1}\frac{1}{2}[/tex]

z = -4

b) (0, 8, 15)

[tex]r=\sqrt{0^{2}+8^{2}}[/tex] = 8

[tex]\theta = \frac{\pi}{2}[/tex]

z = 15

c) (√2,1,1)

[tex]r=\sqrt{(\sqrt{2} )^{2}+1^{2}}[/tex] = [tex]\sqrt{3}[/tex]

[tex]\theta = \frac{\pi}{3}[/tex]

z = 1

d) (−2√3,−2,3)

[tex]r=\sqrt{(-2\sqrt{3} )^{2}+(-2)^{2}}[/tex] = 4

[tex]\theta = \frac{7\pi}{6}[/tex]

z = 3

Answer:

There are 6566 ways to choose 22 croissants with at least one plain croissant, at least two cherry croissants, at least three chocolate croissants, at least one almond croissant, at least two apple croissants, and no more than three broccoli croissants.

Step-by-step explanation:

Given:

There are 5 types of croissants:

plain croissants

cherry croissants

chocolate croissants

almond croissant

apple croissants

broccoli croissants

To find:

to choose 22 croissants with:

at least one plain croissant

at least two cherry croissants

at least three chocolate croissants

at least one almond croissant

at least two apple croissants

no more than three broccoli croissants

Solution:

First we select

At least one plain croissant to lets say we first select 1 plain croissant, 2 cherry croissants, 3 chocolate croissants, 1 almond croissant, 2 apple croissants

So

1 + 2 + 3 + 1 + 2  = 9

Total croissants = 22  

So 9 croissants are already selected and 13 remaining croissants are still needed to be selected as 22-9 = 13, without selecting more than three broccoli croissants.

n = 5

r = 13

C(n + r - 1, r)

= C(5 + 13 - 1, 13)

= C(17,13)

[tex]=\frac{17! }{13!(17-13)!}[/tex]

= 355687428096000 / 6227020800 ( 24 )

= 355687428096000 / 149448499200

= 2380

C(17,13) = 2380

C(n + r - 1, r)

= C(5 + 12 - 1, 12)

= C(16,12)

[tex]=\frac{16! }{12!(16-12)!}[/tex]

= 20922789888000 / 479001600 ( 24 )

= 20922789888000  / 11496038400

= 1820

C(16,12) = 1820

C(n + r - 1, r)

= C(5 + 11 - 1, 11)

= C(15,11)

[tex]=\frac{15! }{11!(15-11)!}[/tex]

= 1307674368000 / 39916800 (24)

= 1307674368000 / 958003200

= 1307674368000 / 958003200

= 1365

C(15,11) = 1365

C(n + r - 1, r)

= C(5 + 10 - 1, 10)

= C(14,10)

[tex]=\frac{14! }{10!(14-10)!}[/tex]

= 87178291200 / 3628800 ( 24 )

= 87178291200 / 87091200

= 1001

C(14,10) = 1001

Adding them:

2380 + 1820 + 1365 + 1001 = 6566 ways

Answer:

The hypotenuse is the longest side in a triangle.

a^2=b^2+c^2.

14^2=9^2+c^2.

c^2=196-81.

c^2=115.

c=√115.

c=10.72~11cm

Answer: Given a sample of 200, we are 90% confident that the true proportion of people who watched educational TV is between 72.1% and 81.9%.

Step-by-step explanation:

[tex]\frac{154}{200} =0.77[/tex]

[tex]1-0.77=0.23[/tex]

[tex]\sqrt{\frac{(0.77)(0.23)}{200} }[/tex]=0.049

0.77±0.049< 0.819, 0.721

Answer:

x = 1, y = -1

Step-by-step explanation:

If we have the two equations:

[tex]4x+3y=1[/tex] and [tex]-3x - 6y = 3[/tex], we can look at which variable will be easiest to eliminate.

[tex]y[/tex] looks like it might be easy to get rid of, we just have to multiply [tex]4x+3y=1[/tex]  by 2 and y is gone (as -6y + 6y = 0).

So let's multiply the equation [tex]4x+3y=1[/tex]  by 2.

[tex]2(4x + 3y = 1)\\8x + 6y = 2[/tex]

Now we can add these equations

[tex]8x + 6y = 2\\-3x-6y=3\\[/tex]

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

[tex]5x = 5[/tex]

Dividing both sides by 5, we get [tex]x = 1[/tex].

Now we can substitute x into an equation to find y.

[tex]4(1) + 3y = 1\\4 + 3y = 1\\3y = -3\\y = -1[/tex]

Hope this helped!

Answer:

a ≈ 1.59

b ≈ 6.69

Step-by-step explanation:

Law of Sines: [tex]\frac{a}{sinA} =\frac{b}{sinB} =\frac{c}{sinC}[/tex]

Step 1: Find c using Law of Sines

[tex]\frac{6}{sin58} =\frac{c}{sin13}[/tex]

[tex]c = sin13(\frac{6}{sin58})[/tex]

c = 1.59154

Step 2: Find a using Law of Sines

[tex]\frac{6}{sin58} =\frac{a}{sin109}[/tex]

[tex]a = sin109(\frac{6}{sin58} )[/tex]

a = 6.68961

Answer:

Size of |E n B| = 2

Size of |B| = 1

Step-by-step explanation:

I'll assume both die are 6 sides

Given

Blue die and Red Die

Required

Sizes of sets

- [tex]|E\ n\ B|[/tex]

- [tex]|B|[/tex]

The question stated the following;

B = Event that blue die comes up with 6

E = Event that both dice come even

So first; we'll list out the sample space of both events

[tex]B = \{6\}[/tex]

[tex]E = \{2,4,6\}[/tex]

Calculating the size of |E n B|

[tex]|E n B| = \{2,4,6\}\ n\ \{6\}[/tex]

[tex]|E n B| = \{2,4,6\}[/tex]

The size = 3 because it contains 3 possible outcomes

Calculating the size of |B|

[tex]B = \{6\}[/tex]

The size = 1 because it contains 1 possible outcome

Answer:

[tex]\boxed{Probability=\frac{1}{2} }[/tex]

Step-by-step explanation:

The probability of flipping at least 1 head from flipping 2 coins is:

=> Total sides of the coins = 4

=> Sides which are head = 2

=> Probability = 2/4 = 1/2

Answer:

y = 4x + 21

Step-by-step explanation:

Hello!

Point-slope form is y - y1 = m(x - x1)

y1 is the y point

x1 is the x point

m is the slope

Put in what you know

y - -3 = 4(x - -6)

Subtracting a negative is the same as adding

y + 3 = 4(x + 6)

Distribute the 4

y + 3 = 4x + 24

Subtract 3 from both sides

y = 4x + 21

The answer is y = 4x + 21

Hope this helps!

Answer:

65 ft

Step-by-step explanation:

The area of a rectangle is

A = lw

6045 = 93*w

Divide each side by 93

6045/93 = 93w/93

65 =w

Answer:

[tex]\huge \boxed{\mathrm{65 \ feet}}[/tex]

Step-by-step explanation:

The area of a rectangle formula is given as,

[tex]\mathrm{area = length \times width}[/tex]

The area and length are given.

[tex]6045=93 \times w[/tex]

Solve for w.

Divide both sides by 93.

[tex]65=w[/tex]

The width of the rectangular garden is 65 feet.

Answer: 3.41x10^3

Step-by-step explanation:

At the beginning of the year, we have:

R = 6.2x10 rats.

And we know that, in one year, each rat produces:

O = 5.5x10 offsprins.

Then each one of the 6.2x10 initial rats will produce 5.5x10 offsprings in one year, then after one year we have a total of:

(6.2x10)*(5.5x10) = (6.2*5.5)x(10*10) = 34.1x10^2

and we can write:

34.1 = 3.41x10

then: 34.1x10^2 = 3.41x10^3

So after one year, the average number of rats is:  3.41x10^3

Answer:

255.8

Step-by-step explanation:

first

1/6*1535

=255.8

Answer:

(a) P(E∪F)= 0.8

(b) P(Ec)= 0.4

(c) P(Fc)= 0.7

(d) P(Ec∩F)= 0.8

Step-by-step explanation:

(a) It is called a union of two events A and B, and A ∪ B (read as "A union B") is designated to the event formed by all the elements of A and all of B. The event A∪B occurs when they do A or B or both.

If the events are not mutually exclusive, the union of A and B is the sum of the probabilities of the events together, from which the probability of the intersection of the events will be subtracted:

P(A∪B) = P(A) + P(B) - P(A∩B)

In this case:

P(E∪F)= P(E) + P(F) - P(E∩F)

Being P(E) = 0.6, P(F) = 0.3 and P(E ∩ F) = 0.1

P(E∪F)= 0.6 + 0.3 - 0.1

P(E∪F)= 0.8

(b)  The complement of an event A is defined as the set that contains all the elements of the sample space that do not belong to A.  The Complementary Rule establishes that the sum of the probabilities of an event and its complement must be equal to 1. So, if P (A) is the probability that an event A occurs, then the probability that A does NOT occur is  P (Ac) = 1- P (A)

In this case: P(Ec)= 1 - P(E)

Then: P(Ec)= 1 - 0.6

P(Ec)= 0.4

(c) In this case: P(Fc)= 1 - P(F)

Then: P(Fc)= 1 - 0.3

P(Fc)= 0.7

(d)  The intersection of two events A and B, designated as A ∩ B (read as "A intersection B") is the event formed by the elements that belong simultaneously to A and B. The event A ∩ B occurs when A and B do at once.

As mentioned, the complementary rule states that the sum of the probabilities of an event and its complement must equal 1. Then:

P(Ec intersection F) + P(E intersection F) = P(F)

P(Ec intersection F) + 0.1 = 0.3

P(Ec intersection F)= 0.2

Being:

P(Ec∪F)= P(Ec) + P(F) - P(Ec∩F)

you get:

P(Ec∩F)= P(Ec) + P(F) - P(Ec∪F)

So:

P(Ec∩F)= 0.4 + 0.3 - 0.2

P(Ec∩F)= 0.8

Answer:

2 * (x + 2) = 50

Step-by-step explanation:

Let's call the unknown number x. "A number and 2" means that we need to add the numbers, therefore it would be x + 2. "Twice" means 2 times a quantity so "twice a number and 2" would be 2 * (x + 2). "Is" denotes that we need to use the "=" sign and because 50 comes after "is", we know that 50 goes on the right side of the "=" so the final answer is 2 * (x + 2) = 50.

Step-by-step explanation:

a.

[tex]s \: = \frac{k}{g} [/tex]

[tex]8 = \frac{k}{1.5} [/tex]

[tex]k \: = 1.5 \times 8 = 12[/tex]

[tex]s = \frac{12}{g} [/tex]

b.

[tex]s = \frac{12}{3} [/tex]

s = 4

Answer:

Solution : Option D

Step-by-step explanation:

The first thing we want to do here is isolate the cos(t) and sin(t) for both the equations --- ( 1 )

x = - 3cos(t) ⇒ x / - 3 = cos(t)

y = 4sin(t) ⇒ y / 4 = sin(t)

Let's square both equations now. Remember that cos²t + sin²t = 1. Therefore, we can now add both equations after squaring them --- ( 2 )

( x / - 3 )² = cos²(t)

+ ( y / 4 )² = sin²(t)

_____________

x² / 9 + y² / 16 = 1

Remember that addition indicates that the conic will be an ellipse. Therefore your solution is option d.

Answer:

a) Compatibility of Equality with Addition, b) Compatibility of Equality with Multiplication

Step-by-step explanation:

a) This expression can be solved by using the Compatibility of Equality with Addition, that is:

1) [tex]3+x = 27[/tex] Given

2) [tex]x+3 = 27[/tex] Commutative property

3) [tex](x + 3)+(-3) = 27 +(-3)[/tex] Compatibility of Equality with Addition

4) [tex]x + [3+(-3)] = 27+(-3)[/tex] Associative property

5) [tex]x + 0 = 27-3[/tex] Existence of Additive Inverse/Definition of subtraction

6) [tex]x=24[/tex] Modulative property/Subtraction/Result.

b) This expression can be solved by using the Compatibility of Equality with Multiplication, that is:

1) [tex]\frac{x}{6} = 5[/tex] Given

2) [tex](6)^{-1}\cdot x = 5[/tex] Definition of division

3) [tex]6\cdot [(6)^{-1}\cdot x] = 5 \cdot 6[/tex] Compatibility of Equality with Multiplication

4) [tex][6\cdot (6)^{-1}]\cdot x = 30[/tex] Associative property

5) [tex]1\cdot x = 30[/tex] Existence of multiplicative inverse

6) [tex]x = 30[/tex] Modulative property/Result

Answer:

3 + x = 27

✔ subtraction property of equality with 3

x over 6  = 5

✔ multiplication property of equality with 6