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Coefficent of x= x* a constant number or a constant number *x

So there is a x and 14.So we multiply and get 14x.There is a -23x and a 1 so we multiply and get 

14x+(-23x).Which equals -9x.

Hope this Helps

The sum of the angles of the quadrilateral = 360 degrees ....

so 

 q   +    4q+17     +   8q+21     +    15q+148 = 360 

28q + 186 = 360 

q = 6.214               Then the largest angle is  15 (6.214) + 148 = 241.2 degrees 

Analyzing the Equation and the Range

Understanding the Equation:

We can factor the equation as follows:

n = ab + 2a + 3b

n = a(b + 2) + 3b

n = a(b + 2) + 3(b + 2) - 6

n = (a + 3)(b + 2) - 6

Implications of the Equation:

This means that n + 6 must be a product of two integers greater than 2.

Considering the Range:

We're looking for n between 70 and 90, inclusive.

So, n + 6 will be between 76 and 96, inclusive.

Finding Suitable Pairs

We need to find pairs of integers greater than 2 whose product lies between 76 and 96.

Lower Bound: The smallest product we can form with integers greater than 2 is 3 * 4 = 12. This is far below 76.

Upper Bound: We can start checking larger products.

5 * 16 = 80

6 * 13 = 78

7 * 12 = 84

8 * 11 = 88

9 * 10 = 90

Counting Valid Values of n:

For each of the products above, we can find a corresponding n by subtracting 6.

Therefore, there are 5 possible values of n between 70 and 90 that can be expressed in the given form.

Answer: There are 5 integers n with 70 ≤ n ≤ 90 that can be written as n = ab + 2a + 3b for at least one ordered pair of positive integers (a, b).

First, let's note that

\(3 + 4 + 5 = 3 *4\)

\(13 + 14 + 15 = 3 * 14\)

This pattern will keep going. Thus, we just have

\(3 [ 4 + 14 + 24 + ....... + 994 ]\)

No. of terms =  100

Thus, our final answer is

\(3 [ 994 + 4 ] [ 100 / 2 ] = 149,700\)

Thanks! :)

First, let's calculate the distance Will travels before grace even leaves. 

Since he traveled for 45 minutes, we have

\((45 min)(50 m/min) = 2250 m\)

So when Grace starts, she and Will have

\((2800 m) – (2250 m) = 550 m\) to cover. 

Let's let t be the amount of time it takes for the two to catch up to each other after Grace starts rowing. 

We have the equation

\((50)(t) + (30)(t) = 550 \\ 80t = 550 \\ t = 550/80 = 6.875\)

This is approximately 6 minutes and 52 seconds. 

So the clock time they meet is 6 min 52 sec after Grace starts.

Grace started at 2:45, so add 6 min 52 sec and the clock will read 2:51:52 pm when they meet  

Thus, our final answer is \(2:51:52\)

Thanks! :)

Alright. Let's first put y in terms of x. 

We get \(y=25-x\). 

Subbing this value back into the second equation, we get 

\(3x+75\). 

Since x can't be negative, the smallest possible value of x is 0. 

When x is 0, we have \(3(0)+75=75\)

So 75 is our final answer. 

Thanks! :)

First of all, let's focus on the second equation. 

We can factor the right side of the equation to get \((x+y)(x^2+xy+y^2)=162+x^2+y^2\)

Since x+y is equal to 10 from the first equation, the equation becomes

\(10(x^2+xy+y^2)=162+x^2+y^2\)

\(10x^2+10xy+10y^2= 162+x^2+y^2\)

\(9x^2+9y^2+10xy=162\)

Now, isolating x in the first equation, we get

\(x=10-y\)

Now plugging this value of x into the second equation, we get

\(9(10-y)^2+9y^2+10(10-y)y=162\\ 9(100-20y+y^2)+9y^2+100y-10y^2=162\\900-180y+9y^2+9y^2+100y-10y^2=162\\ 8y^2-80y+900=162\\8y^2-80y+738=0\)

However, plugging this into the quadratic formula, notice that unofruntately, the descriminant is less than 0, menaing the eventual solutions are NONREAL numbers. 

Thanks! :)

First, we can write a handy equation to solve for the constant and find the 3 terms. 

Let's let the constant added to every number be x. 

Since the three terms for a geometric series, we have the equation

\(\frac{100+x}{60+x} = \frac{140+x}{100+x}\)

Now, when we corssmultiply and then expand everything out, we get

\((x+100)(100+x)=(x+60)(140+x)\\ 200x+x^{2}+10000=140x+x^{2}+{8400+60x}\\ 200x+x^{2}+10000=200x+x^{2}+8400\)

Now, we bring all terms to one side of the equation. We have

\(200x+x^{2}+10000-x^{2}-200x-8400=0\\200x+1600-200x=0 \\ 1600=0\)

However, this statement is obviously not true, meaning that x is invalid. 

This also means that there are NO solutions to this given problem. 

*Note, I may have made a mistake. Not sure. 

Thanks! :)

If angle XWZ = 60°, angle WXY = 180° - 60° = 120° because XY and WZ are parallels.

However, this means that Angle ZXW must be less than 120°.

In the problem though, it clearly defines ZXW as 120, meaning this problem is invalid. 

Thanks! :)

Hi! Although I don't have much expertience witht he calculator, I would reccomend websites like 

Symbolab or Mathway

for the job. Although their explenations are simple, their solutions are almost always correct. 

I wouldn't reccomend it, but sometimes AI bots such as chatgpt also come up with the correct solution, although MUCH more inaccurate. 

Also, it wouldn't hurt to solve the problem yourself to verify which website is most accurate! :)

I hope this helped!

Thanks! :)

Alright, first let's note that we can caluclate the number of distinct arrangements through the equation

\(\text{Number of Distinct Arrangements} = \frac{n!}{n_1! \cdot n_2! \cdot n_3! ... \cdot n_k!}\)

where n is the total number of items to arrange, and n1,n2,…,nk are the frequencies of the distinct items (for example, i in mississippi appears 4 times). 

Thus, let's list our the letters we have in the word. 

Mississippi has 11 letters, so n is 11. 

In the word, i appears 4 times, m appears once, s appears twice, and p appears twice. 

Thus, plugging in all these numbers, we have

\(\frac{11!}{1! \cdot 4! \cdot 2! \cdot 21} = 34650\)

So 34650 is our answer. 

Thanks! :)

Base1 = 12 units

Base 2 = 12 + 16 = 28  units

median = (12 + 28)/2 = 20 units 

We can use the Law of Cosines to solve this problem. Applying it, the law states that

\(AE^2 = AF^2 + FE^2 - 2 (AF * FE) cos (60°) \)

Now, plugging in the values from the problem we already know, we find that

\( AE^2 = 2^2 + 5^2 - 2(2* 5) (1/2)\\ AE^2 = 19\\ AE = \sqrt {19}\)

Thanks! :)