Solve the system. xy = 4?

Question

4/x + 1/y = 2

Answer 1

xy = 4

4/x + 1/y = 2

Consider the first equation:
y = 4/x. Substitute this in the second equation:

4/x + 1/4/x = 2 which then gives,

4/x + x/4 = 2
Now rewrite as

4/x + x/4 = 2/1 Multiply through by 4x (the LCM of the equation). This gives:

16 + x^2 = 8x
that is x^2 - 8x + 16 =0
factorise to get : (x - 4)(x - 4) = 0
therefore, x = 4.

Now substitute the value of x in the first equation:

4y = 4
y = 4/4 = 1 (divide thru by 4, the co - efficient of y)

Therefore, x = 4, y =1.

Answer 2

xy = 4 y= 4/x replace in equation a million x^2 + (4/x)^2 = 8 x^2 + sixteen/x^2 = 8 x^2/a million + sixteen/x^2 = 8 Fractions - make it the comparable denominator (x^2)(x^2)/x^2 + sixteen/x^2 = 8 (x^4 + sixteen)/x^2 = 8 x^4 + sixteen = 8 (x^2) x^4 + sixteen = 8x^2 sixteen = 8x^2 - x^4 sixteen = x^2 (8 - x^2) So answer would be sixteen = x^2 x=4 or -4 OR sixteen = 8 - x^2 x^2 = 8 - sixteen x^2 = - 8 x = sq. root of -8 ---> No answer. So the respond is the 1st selection the place x= 4 or - 4 If x = 4, xy = 4 (4)y = 4 y = a million If x = - 4 xy = 4 (-4) y = 4 y = -a million

Answer 3

xy = 4
4/x + 1/y = 2

This is one of those rare problems where the answer just pops out

4/x + 1/y = 2
1 + 1 = 2

x = 4, y = 1

Or take the more rigorous approach which your instructor would prefer.

Use substitution

xy = 4
y = 4/x

Substitute into 2nd equation

4/x + 1/(4/x) = 2
4/x + x/4 = 2
multiply thru by 4x

16 + x² = 8x

x² - 8x +16 = 0

(x - 4)(x - 4) = 0

x = 4

xy = 4

4y = 4

so y = 1
.

Answer 4

xy = 4
4/x + 1/y = 2

multiply the second equation by xy, which is the LCD.

4y + x = 2xy

since xy = 4 and y =x/4

4x/4 + x = 2(4)
x + x = 8
2x = 8
x =4

y = 4/4 = 1

Answer 5

xy=4
y=4/x

4/x + x/4 = 2

(16+x^2)/4x = 2
x^2 + 16 = 8x
x^2-8x+16=0
(x-4)^2=0
x=4
y=4/x=1

x=4
y=1

Answer 6

xy = 4 (eq. 1)
4/x + 1/y = 2 (eq. 2)
________________
xy = 4
x = 4/y (eq. 3)
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Subtitute eq. 3 in eq. 2
4/4/y + 1/y = 2
y + 1/y = 2
Multiply both sides by y
y( y + 1/y = 2)
y^2 + 1 = 2y
y^2 - 2y + 1 = 0
(y - 1) (y - 1) = 0
y -1 = 0
y = 1 ==> value of y
___________________
Substitute the value of y in eq. 1 to find x
x(1) = 4
x = 4/1
x = 4 ==> value of x
_________________
To check:
xy = 4 (eq. 1)
4 ( 1) = 4
4 = 4
_______________
4/x + 1/y = 2 (eq. 2)
4/4 + 1/1 = 2
1 + 1 = 2
2 = 2

Answer 7

x = 4
y = 1