Please solve x and y from this equivalence : x^2 - y^2 + 1 = xy - 5y , x+1 = y+2. Thanks?

Answer 1

x + 1 = y + 2
x = y + 1

x^2 - y^2 + 1 = xy - 5y
(y + 1)^2 - y^2 + 1 = (y + 1)y - 5y
y^2 + 2y + 1 - y^2 + 1 = y^2 + y - 5y
2y + 2 = y^2 - 4y
y^2 - 6y - 2 = 0

y = (6 +- sqrt((-6)^2) - (4*1*-2)))/2
= (6 +- sqrt(36 + 8))/2
= (6 +- sqrt(44))/2
= (6 +- 6.6)/2

y = 12.6/2 or -0.6/2

y = 6.3 or -0.3
x = 7.3 or 0.7

Answer 2

x^2 - y^2 + 1 = xy - 5y

The simplest way to solve this is to just solve for x + 1 = y + 2 for one of the variables. That is,
x + 1 = y + 2 means that
x = y + 1.
So we plug in every occurrance of x with y + 1.

(y + 1)^2 - y^2 + 1 = (y+1)y - 5y

And then we expand.

y^2 + 2y + 1 - y^2 + 1 = y^2 + y - 5y

And simplify

2y + 2 = y^2 - 4y

Bring everything over to the right hand side, to get
0 = y^2 - 4y - 2y - 2
OR
0 = y^2 - 6y - 2

This is a quadratic, and we can use the quadratic formula but I'd prefer to complete the square

0 = y^2 - 6y + 9 - 2 - 9
0 = y^2 - 6y + 9 - 11
0 = (y - 3)^2 - 11
11 = (y - 3)^2
Therefore,
y - 3 = +/- sqrt(11), and
y = 3 +/- sqrt(11).

So we have two values for y:
y = 3 + sqrt(11) and y = 3 - sqrt(11).

To solve for x, we just plug in each value of y for the linear equation x + 1 = y + 2 (or, x = y + 1)

Let y = 3 + sqrt(11). Then
x = y + 1 = [3 + sqrt(11)] + 1 = 4 + sqrt(11)

Let y = 3 - sqrt(11). Then
x = y + 1 = [3 - sqrt(11)] + 1 = 4 - sqrt(11)

Therefore, we have the solutions

x = 3 + sqrt(11), y = 4 + sqrt(11)

x = 3 - sqrt(11), y = 4 - sqrt(11)

Answer 3

x^2 – y^2+1=xy-5y, x+1=y+2
x=y+2-1=y+1


(y+1)^2 – y^2+1=(y+1)y – 5y
y^2+2y+1- y^2+1=y^2+y-5y

2y+2= y^2 -4y

y^2-6y=2
I think that should be enough for you to continue

Answer 4

if.... x+1=y+2
then x=y+1

substitutue y+1 for every x in the other equation

(y+1)^2-y^2+1=(y+1)y-5y
y^2+2x+1-y^2+=y^2+y-5y
2y+2=y^2+y-5y
0=y^2-6y-2

then slove for y by using the quadratic equation

sub in the values for a,b,c

y=-b+-(sqrt b^2-4ac)/2a

if 0=ay^2+by+c

sub in and your answers for y should come as 6.3 and -0.3

so there for x sub back into the original problem and x is

either 7.3 or 0.7

Answer 5

x+1 = y+2
x=y+1

(y+1)^2+1=(y+1)y-5y
y^2+y+y+1+1=y^2+y-5y
y^2+2y+2=y^2-4y
6y=-2
y=-2/6
y= -1/3