Find all ordered pairs (x, y) that satisfy the system of equations

[hancyu]

1. Find all ordered pairs (x, y) that satisfy the system of equations given below:
* (x + 1)^2 + |(x + 1)(y − 2)| + (y − 2)^2 = 7
* x^2 + y^2 + 2x − 4y − 5 = 0

2. If the area of a certain rectangle is 48, and the length of its diagonal is 10, find its perimeter.

here's what i did.

1. 2nd equation is equal to "(x+1)^2+(y-2)^2=10"
what should i do next?

2. i know that the answer to this is 28 because the sides are equal to 6 & 8... but how do i solve for it?

 

[statdad]

The second equation is the 'standard form' of a shape you should be familiar with at this time.

I'm not sure how you found the answer to 3, since you don't have any work to present (trial and error?)

Try drawing a rectangle, one side labeled $$x$$, the other $$y$$. Since the area is $$48$$, you know that $$xy = 48$$ so

$$y = \frac{48} x$$

Now substitute into the formula for finding the known length of the diagonal and solve.

For your first question, I have a question. Did you intend to type

$$(x+1)^2 + (x+1)(y-2) + (y-2)^2 = 7$$

or was it what I have below, with the middle term in absolute values? (this is what seems to appear)

$$(x+1)^2 + |(x+1)(y-2)| + (y-2)^2 = 7$$

I'm guessing there aren't any absolute value signs in the original problem.

 

[hancyu]

actually, there are absolute value signs.

 

[hancyu]

so... in the 2nd question.

(48/x)^2 + x^2 = 100

2304/(w^2) + (w^2) = 100

[2304 + (w^4)]/ (w^2) = 100

what should i do to get rid of those exponents?

 

[HallsofIvy]

1. Find all ordered pairs (x, y) that satisfy the system of equations given below:
* (x + 1)^2 + |(x + 1)(y − 2)| + (y − 2)^2 = 7
* x^2 + y^2 + 2x − 4y − 5 = 0
2. If the area of a certain rectangle is 48, and the length of its diagonal is 10, find its perimeter.

here's what i did.

1. 2nd equation is equal to "(x+1)^2+(y-2)^2=10"
what should i do next?

From the second equation you know, as you say, that (x+1)²+ (y-2)²= 10. You can replace those same terms in the first equation by 10:
(x+1)²+ |(x+1)(y-2)|+ (y-2)²= 7 becomes
|(x+1)(y-2)|+ 10= 7 or |(x+1)(y-2)|= -3. Now, what values of x and y satify that?

2. i know that the answer to this is 28 because the sides are equal to 6 & 8... but how do i solve for it?

If the sides have length x and y, then you know that xy= 48 and x²+ y²= 10. You need to solve those two equations. From the first, y= 48/x so the second becomes x²+ 48²/x²= 10. Multiply both sides of the equation by x² and you get a quadratic equation for x².

 

[HallsofIvy]

so... in the 2nd question.

(48/x)^2 + x^2 = 100

2304/(w^2) + (w^2) = 100

[2304 + (w^4)]/ (w^2) = 100

what should i do to get rid of those exponents?

Multiply both sides of the equation to get 2304+ w⁴= 100w² which is the same as (w²)²- 100(w²)+ 2304= 0, a quadratic equation in w².

 

[hancyu]

From the second equation you know, as you say, that (x+1)²+ (y-2)²= 10. You can replace those same terms in the first equation by 10:
(x+1)²+ |(x+1)(y-2)|+ (y-2)²= 7 becomes
|(x+1)(y-2)|+ 10= 7 or |(x+1)(y-2)|= -3. Now, what values of x and y satify that?

none? coz absolute values can nver be negative,ryt?

If the sides have length x and y, then you know that xy= 48 and x²+ y²= 10. You need to solve those two equations. From the first, y= 48/x so the second becomes x²+ 48²/x²= 10. Multiply both sides of the equation by x² and you get a quadratic equation for x².

so x^2 = 64

x= 8

y = 6?

 

[hancyu]

3. Find the domain of the following functions:
(a) f(x) =
(x^2 − 1)^(1/3) − 1 / [|3-x| - (x-3)^(1/3) - 3]

(b) f(x) =
(x^2 − 2x + 1)^(1/2) / [|2x − 3| + 1]

(c) f(x) = x^3 / (|3 − x| − (x – 3)^(1/3))

how do i do these?

 

[statdad]

The domain of a rational function depends only on the denominator, as long as there are no radicals or even roots in the numerator.
The domain for each can be found by determining all numbers (if any) that make the denominator zero: once those values are eliminated, the domain is all values that remain.

 

[hancyu]

for a. x>12

b. all real #s

c.how do i do this?

 

[HallsofIvy]

for a. x>12

b. all real #s

c.how do i do this?

3. Find the domain of the following functions:
(a) f(x) =
(x^2 − 1)^(1/3) − 1 / [|3-x| - (x-3)^(1/3) - 3]

(b) f(x) =
(x^2 − 2x + 1)^(1/2) / [|2x − 3| + 1]

(c) f(x) = x^3 / (|3 − x| − (x – 3)^(1/3))

how do i do these?

What values of x make |3- x|- (x-3)^1/3= 0?

 

[hancyu]

What values of x make |3- x|- (x-3)^1/3= 0?

is the answer 3? how about the other numbers, are they correct?

 

[HallsofIvy]

I asked a question you are supposed to ANSWER, not ask another question!

|3-x|- (x-3)^1/3= 0 is the same as |x- 3|= (x-3)^1/3.

Now, suppose x> 3. then |x-3|= x- 3= (x-3)^1/3. Can you solve that?

If x< 3, then |x-3|= 3- x= (x-3)^1/3.