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\(4-x\ge3\); \(3x+5\ge2\)

In a rectangular coordinate system, the line $5y = 2x$ intersects the line $3x - 4y = 7$ at point $Z$. What is the sum of the coordinates of point $Z$?

The points $(x, y)$ represented in this table lie on a straight line. The point $(28, t)$ lies on the same line. What is the value of $t?$

When you add WHAT? the harmonic series up to 1 over p-1? What reduced fraction are you talking about?

(a + b)(a + b)  =  (a)(a) + (a)(b) + (b)(a) + (b)(b)  =  a² + ab + ab + b²  =  a² + 2ab + b²

So....   a² + 2ab + b²  =  (a + b)(a + b)

And if

(a² + 2ab + b²)(a + b)  =  125      then....

(a + b)(a + b)(a + b)  =  125

(a + b)³  =  125

a + b  =  \(\sqrt[3]{125}\)

a + b  =  5

Let's check your factoring by multiplying out  (x - 6)(x + 2) :

(x - 6)(x + 2)  =  (x)(x) + (x)(2) + (-6)(x) + (-6)(2)

                      =  x² + 2x - 6x - 12

                      =  x² - 4x - 12

 x² - 4x - 12   is not the same as  3x² - 4x - 4  ,  so

(x - 6)(x + 2)  is not the same as  3x² - 4x - 4

We can factor  3x² - 4x - 4  like this....

So   3x² - 4x - 4  =  (x - 2)(3x + 2)

To find the  x  values that make the expression equal to zero, set  (x - 2)(3x + 2)  equal to  0 .

(x - 2)(3x + 2)  =  0      Set each factor equal to zero and solve for  x .

x - 2  =  0       or       3x + 2  =  0

   x  =  2         or          x  =  -2/3

So the  x  values that make  3x² - 4x - 4  equal zero are  x = 2  and  x = -2/3 .

I have been asked why 1,2,3,4,5 and 6 are not also answers.

The mod of a number is the remainder that you get when you divide by the modulus.

If the mod is less then 7 then the remainder cannot be 7 or more.

127=0(mod1)

127=1(mod2)

127=1 (mod3)

127=3(mod4)

127=2(mod5)

127=1(mod6)

127=7(mod8)    Bingo, this is the smallest.

x^{1 \over3}=4{1 \over x^3}

\(x^{1 \over3}=4{1 \over x^3}\\ x^{1 \over3}=4x^{-3}\\ x=4^3*x^{-9}\\ x^{10}=4^3\\ x=4^{3/10}\\ x=2^{3/5}\\ x=\sqrt[5]{8}\\ x\approx 1.5157 \)

this is the only positive real answer. 

I think our guest's negative answer also works.  It may not work with convention though

The cube root is in the question. so is a negative allowed.. I am not sure.

I think there are other complex answers as well.

I put in the functions in desmos, and here is what I got:

https://www.desmos.com/calculator/obzfylom2j

I split the trianlge into two, and calculated the summation of their areas.

(10 - 4)*2 + (10 - 4)*12 = 84

The area of the triangle is 84.

I hope this helped,

Gavin.

Hey supermanaccz!

If \(127 \equiv 7 \pmod{n}\), then n is a divisor of 127 - 7 = 120.

The prime factorization of 120 is \(2^3 \cdot 3 \cdot 5\)

which has \((3 + 1)(1 + 1)(1 + 1) = 16\) positive divisors.

Therefore, there are 16 possible values of n.

I hope this helps,

Gavin.

There was the same question 1 day ago, https://web2.0calc.com/questions/how-many-positive-integers-n-satisfy-127-equiv-7

Hey RB!

If \(127 \equiv 7 \pmod{n}\), then n is a divisor of 127 - 7 = 120.

The prime factorization of 120 is \(2^3 \cdot 3 \cdot 5\)

which has \((3 + 1)(1 + 1)(1 + 1) = 16\) positive divisors.

Therefore, there are 16 possible values of n.

I hope this helps,

Gavin.

Can somebody translate this into proper LaTex?

Tried, but didn't work!!. Sorry!.

When you add everything up, you reduce the fraction,

Would you like the solution?

To find the total amount of people, we use an algorithm.

59 + 37 + 54 - 25 - 12 - 28 + 10 = 95.

There are 95 people who have any type of fruit at all.

This means that we can find the amount of people without any fruit. 

100 - 95 = 5

There are 5 people with no fruit.

I hope this helped,

Gavin

The problem asks for consecutive intergers. 

10 + 11 = 21

The largest posible number of positve, consecutive intergers must start with 1, since it is the smallest positive integer.

1 + 2 + 3 + 4 + 5 + 6 = 21

This is the largest amount of consecutive integers. 

6 is your answer. 

I hope this helped,

Gavin.

20+1?

The coordinates of the vertices of triangles are listed in the options below. Which is a right triangle?

GYanggg, what reduced fraction are you talking about in your second question?

divide maybe?

Hey there!

We need to factor out as much as we can in your cube root.

After we do, we have:

-4 times the cube root of 2.

I hope this helped,

Gavin. 

Your welcome!

Did you have the same question too?

I think the scale on your map should be 1 1/3 inches per 8 miles. If that is the case, then:

8 / (4/3)  x  8 = 48 miles - the distance between the two cities.