Guinevere

first, re-write the equation in LaTeX.

${1\over2}y+{1\over4}=3+\boxed{\phantom{400}}y$

Instead of a box, substitute it for x.

${1\over2}y+{1\over4}=3+xy$

To get rid of the fractions, multiply both sides by 4.

$4*({1\over2}y+{1\over4})=(3+xy)*4$

simplified...

$2y+1=12+4xy$

move all numbers to one side

$(2y+1)-(2y+1)=12+4xy-(2y+1)$

simplified...

$4xy-2y+11=0$

the easiest way to get more than one solution is to make the equation a binomial.

to make $4xy-2y+11=0$ a binomial, we can just say that x=y

simplified...

$4y^2-2y+11=0$

since x=y, y must be placed in the box.

The only thing I am worried about is that it might not want a variable for the box, but instead an actual number.

if $185=100% of the price, and a 5% markup is added, we add 5% to the total %.
105% is the same as 1.05x.

1.05*185=$194.25

The line y = 1 is a horizontal line that is one above the x-axis.

Since the x-coordinate determines how far left or right a point is, it stays the same at y=1.

To reflect a point is to mirror it accros a line. Since -5 is 6 away from 1, the reflected point must also be 6 away from 1. That point is 7.

Putting the coordinates together, we get the coordinate (6, 7), of course with the x-coordinate stated first.

We can start witht the number 500 and count all of the squares/cubes under that.

22² (484) is the closest we can get to 500 without going over it.
7³ (343) os the closest we can get to 500 without going over it.

22+7 is 29, which is how many we got rid of (almost, but not really)

Since 1 is both a square and a cube, we need to subtract 1 for the double count. 
Since 64 is both a square and a cube, we need to subtract another 1 for the double count.

So to compensate for each square and cube not included, we can add 27 (29-2) to 500
The 500th term would be 527, but 512 is 8^3, so we also have to compensate for that.

527+1=528

Please explain how that answer is wrong

(sorry if I am missing something very basic)

(6a³+92a²-7)-(-7a³+5a²-7)

First, distribute the negative sign to the right parenthesis, and take away both parenthesis.

6a³+92a²-7+7a³-5a²+7

Combine like-terms.
13a³+87a²

Pull out an a².

a²(13a+87)

If each place value has to have a 2, 5, or 8, there are 3 possibilities for each place value.

3*3*3= 27

ABA₆+411₆=1215₆

Although there are other ways to do this, I would start by converting the two right values to base ten.

The now converted equation is:

ABA₆+151₁₀=299₁₀

If we then subtract 151 from both sides, we get:

ABA₆=148
We can then use the equation 36A+6B+A=148, which simplifies to 
37A+6B=148

The only non-extraneous values that work for this equation are A=4 and B=0

4-0=4, which makes the answer 4.

If a word consists of _ _ _ blanks, and A and B must be used once each,

there could be 6 different positions. (AB{}, A{}B, BA{}, B{}A, {}AB, {}BA)

the blank has four different values for each position (A, B, C, or D)

If we multiply the number of postions by what value could be filled in for each postion, we get 4x6=24

There are therefore 24 ways.

Would you please explain how you got this answer?

If it takes four painters 1.25 days to finish a job, it would take one person 5 days to finish the same job. If this five days is divided into 3 people, it will take 5/3 days for them to complete the job.

since a two-pound cake requires 1.5 cups of flower, a one-pound cake would take 0.75 cups of flower.

0.75*12=9

$\sqrt(x\sqrt(x))$ on the inside of the parenthesis, move the x to the inside of the other sqrt, so that

$\sqrt(\sqrt(x^3))$ the inside sqrt can be combined with the 3 on the inside, so that
$\sqrt(x^{3/2})$. This is also the answer.

It can also be expressed as $x^{3/4}$

Each hour is 60 minutes, so convert the second value to minutes.

3*60+30=210 (for converting to minutes)

If she can eat 1 popsicle per 15 minutes, divide 210 by 15. This comes out to 14, so she can eat 14 popsicles in 3 hours and 30 minutes.

To simplify radicals such as these, there are multiple methods. One method is to find the prime factorization of the number and edit it from there.

(2*2*2*2*7) is the prime factorization for 112. You can pull out pairs of numbers to the front of the sqrt, although cancel one in the process. this would then be $2*2\sqrt7$ this is then simplified to $4\sqrt7$, which is the answer.

To find the perimeter, we must first find the distance from (1, -2) to  (4, -10). To do this, we can use the distance formula.

$\sqrt((x\delta)^2+(y\delta)^2),$ where $\delta$=the change in positioning for the point.

We can plug in for this equation to get $\sqrt((4-1)^2+(-10--2)^2)$

Simplifying this, we get $\sqrt(3^2+(-8)^2)$, then $\sqrt(9+64)$, and finally $\sqrt73$

Since there are 4 sides to a square, multiply $\sqrt73$ by 4 to account for all sides.

The final answer should be $4\sqrt73$ since 73 is prime.

First, we set each person to a variable.

If Peter=x, or (6/6)x

    Ned=2 1/3 x, or (7/3)x, or (14/6)x,

 Morris= (7/6)x

Since the total number of apples was 999, (6/6)x+(14/6)x+(7/6)x=999

Adding the x coefficients together, we get (27/6)x=999.

To get rid of the fraction, multiply both sides by (6/27).

This results in x=999*(6/27)

999*(6/27)=222

If we plug this in for each person, we get

  Peter=222,

    Ned=518,

Morris=259

It is also helpful to make a graph (as shown below) for convenience:

 (obviously the 3rd column would not be needed for this problem)