Math Prob 27?

Question

How many points do the graphs of y = x^2 and xy = 27 have in common?

Answer 1

Mena's correct, but she didn't show how to get the point.

If xy = 27, then x = 27/y.

Plug that into the first:

y = x² = (27/y)² = 729/y²

Multiply both sides by y²:

y³ = 729

Take cube root of both sides:

y = 9

Solve for x:

x = 27/y = 27/9 = 3

Answer 2

There is yet another way of analyzing this problem, without having to calculate much at all.

Note that y = x^2 is the equation for a parabola which opens upward, since the coefficient of the x^2 term is positive. That means that all real solutions to this must be in either the first or second quadrant of the Cartesian plane.

If xy = 27, then when x is negative, y must also be negative, for obvious reasons. Since this only occurs in the third quadrant, which does not contain any of the solution set for x^2 = 27, then we can throw out the negative value for the solution to x^2 = 27, because it will result in a non-solution to the entire system. Knowing this, we can then calculate what the positive value for x will be, which most of us can probably do in our heads. Then we can find the corresponding positive value for y, which is also a trivial task.

Answer 3

This is another way of saying solve the equations simultaneously.

If the points are in common, both equations are true at the same time.

y = x^2

xy = 27............rearrange this one to givey = 27/x ..(divide both sides by x

now you can say X^2 = 27/x

Multiply both sides by x to get rid of x in the denominator.

x^3 = 27

x = 3 and as y = x^2, y = 9

(Remember cube roots don't play the same trick as square roots, so you don't have to look for a -ve answer.)

There is one point where they touch

(3,9)

Answer 4

I would have to say 2 points because x^2 is a parabala and xy=27 is a line.

Answer 5

1 point (3,9)

Answer 6

(3,9)