I need to find the sum of -1/3+(-4/15)

1) Solve the following system of equations algebraically. Give your solution as two

Bad White [126]

For this case we have the following system of equations:

$y = x ^ 2 + 7x + 12\\y = 2x + 8$

Equating both equations we have:

$x ^ 2 + 7x + 12 = 2x + 8\\x ^ 2 + 7x-2x + 12-8 = 0\\x ^ 2 + 5x + 4 = 0$

We must find the solutions, for this we factor. We look for two numbers that, when multiplied, result in 4 and when added, result in 5. These numbers are 4 and 1:

$4 + 1 = 5\\4 * 1 = 4$

Then, the factorized equation is of the form:

$(x + 4) (x + 1) = 0$

Thus, the solutions are:

$x_ {1} = - 4\\x_ {2} = - 1$

We look for solutions for the variable "y":

$y_ {1} = 2 (-4) + 8 = -8 + 8 = 0\\y_ {2} = 2 (-1) + 8 = -2 + 8 = 6$

Thus, the system solutions are given by: $(x_ {1}, y_ {1}): (- 4,0)\\(x_ {2}, y_ {2}): (- 1,6)$

ANswer:

$(x_ {1}, y_ {1}): (- 4,0)\\(x_ {2}, y_ {2}): (- 1,6)$

8 0

3 years ago

The sequence$$1,2,1,2,2,1,2,2,2,1,2,2,2,2,1,2,2,2,2,2,1,2,\dots$$consists of $1$'s separated by blocks of $2$'s with $n$ $2$'s i

kicyunya [14]

Consider the lengths of consecutive 1-2 blocks.

block 1 - 1, 2 - length 2

block 2 - 1, 2, 2 - length 3

block 3 - 1, 2, 2, 2 - length 4

block 4 - 1, 2, 2, 2, 2 - length 5

and so on.

Recall the formula for the sum of consecutive positive integers,

$\displaystyle \sum_{i=1}^j i = 1 + 2 + 3 + \cdots + j = \frac{j(j+1)}2 \implies \sum_{i=2}^j = \frac{j(j+1) - 2}2$

Now,

$1234 = \dfrac{j(j+1)-2}2 \implies 2470 = j(j+1) \implies j\approx49.2016$

which means that the 1234th term in the sequence occurs somewhere about 1/5 of the way through the 49th 1-2 block.

In the first 48 blocks, the sequence contains 48 copies of 1 and 1 + 2 + 3 + ... + 47 copies of 2, hence they make up a total of

$\displaystyle \sum_{i=1}^48 1 + \sum_{i=1}^{48} i = 48+\frac{48(48+1)}2 = 1224$

numbers, and their sum is

$\displaystyle \sum_{i=1}^{48} 1 + \sum_{i=1}^{48} 2i = 48 + 48(48+1) = 48\times50 = 2400$

This leaves us with the contribution of the first 10 terms in the 49th block, which consist of one 1 and nine 2s with a sum of $1+9\times2=19$ .

So, the sum of the first 1234 terms in the sequence is 2419.

8 0

2 years ago

Quincy deposits $400 in a new bank account that earns 4 simple interest per year. What is the total amount in the account after

Kobotan [32]

Ÿū tj Yates tu hamase de Beas

6 0

3 years ago

Sin(x2 + y2) da r , where r is the region in the first quadrant between the circles with center the origin and radii 2 and 5

sweet [91]

Converting to polar coordinates gives

$\displaystyle\iint_R\sin(x^2+y^2)\,\mathrm dA=\int_{\theta=0}^{\theta=\pi/2}\int_{r=2}^{r=5}\sin(r^2)r\,\mathrm dr\,\mathrm d\theta$
$=\displaystyle\pi\int_{r=2}^{r=5}2r\sin(r^2)\,\mathrm dr$
$=\displaystyle\pi\int_{r^2=4}^{r^2=25}\sin(r^2)\,\mathrm d(r^2)$
$=-\cos(r^2)\bigg|_{r^2=4}^{r^2=25}$
$=\pi(\cos4-\cos25)$

3 0

3 years ago

the perimeter of a rectangle is 62 feet.if the width of the rectangle is 19ft. what is the length of the rectangle

denis23 [38]

Perimeter = 2(l+w) or 2(length+width)

So far you have the width, which is 19feet. Since you have two sides of the rectangle which are 19 feet, add 19+19=38

Subtract 62-38 to get 24.

You now have the length of both of the sides combined. So you need to divide 24/2 = 12
The length of the rectangle = 12ft.

6 0

3 years ago

Read 2 more answers