Simultaneous equations?

Question

How to find x and y if :

Root(y^2-2x)=y-x
and
2^x+3=2^(y+1)

Thank you !

Answer 1

Same way as always: first get y or x in terms of the other.

In this case start with the first equation, and square each side:

y^2 - 2x = y^2 - 2xy + x^2
x^2 + 2x = 2xy
2xy = x(x + 2).

If x = 0 this is true, so save that to look at later. Otherwise, divide by 2x:

y = (x + 2)/2

Now look at equation 2. If x = 0

3 = 2^(y + 1)
log2(3) = y + 1
log2(3) - 1 = y
y = log2(3/2)

Case 2: y = (x + 2)/2
2^x + 3 = 2^(x/2 + 2)
2^(x/2)*2^(x/2) - 4*2^(x/2) + 3 = 0

Quadratic formula: let w = 2^(x/2):
w^2 - 4w + 3 = 0
(w - 3)(w - 1) = 0
w = 3, w = 1
2^(x/2) = 1: x = 0, already have it
2^(x/2) = 3: x = 2log2(3)

You should check the answers whenever you square something, and I may have made a mistake in the numbers but it is basically right.

Answer 2

Use substitution. Solve either equation for x or y...Then substitute and solve the single varable equation.

example : Solve equation 2 for X in terms of Y. Substitute this solution for X into the first equation and you will get an equation that is single variable (only Y in the equation). Solve for Y. Then plug the value for Y into either equation and solve for X.