- #1
Euler_Euclid
- 10
- 0
find all real values such that:
[tex]x+y=1[/tex]
and [tex]xy-z^2=1[/tex]
This one was in one of the exams we had.
[tex]x+y=1[/tex]
and [tex]xy-z^2=1[/tex]
This one was in one of the exams we had.
HallsofIvy said:[tex]x= \cdots= \frac{1\pm \sqrt{8z^2+ 9}}{2}[/tex]
gel said:As x must be positive, that can't be right.
The problem is easy to see graphically. The set of values for (x,y) is just the intersection of the region xy>=1 with the line x + y = 1, which gives a closed line segment.
z2 is nonnegative.Jarle said:How on Earth can you deduce that?
HallsofIvy said:Nothing at all difficult about that, except that I presume you mean "find all 'triples' of real numbers, (x, y, z) that satisfy the equations". From x+ y= 1, y= 1- x. Putting that for y in the second equation, x(1- x)- z2= x- x2- z2= 1 which we can rewrite as the quadratic equation x2- x- (z2+ 1)= 0
The purpose of solving for real values of x, y, and z is to find the specific numerical values that satisfy the given equations. These values can then be used in further calculations or to determine the behavior of a system.
To solve for real values of x, y, and z, you must first rearrange the equations to isolate one variable. Then, substitute the value of the isolated variable into the other equation to find the remaining variables. This process may need to be repeated multiple times to find all three values.
Yes, there can be more than one set of real solutions for x, y, and z. This occurs when the equations have multiple intersections or when one equation is a function of the other. In these cases, there may be infinite solutions or a finite number of solutions.
The condition x+y=1 is a constraint that limits the range of possible solutions for x and y. This means that any solutions found must satisfy this constraint, which can help narrow down the possible values for x and y.
Yes, there are alternative methods for solving these equations, such as using graphical methods or substitution methods. These methods may be more efficient or easier to understand for certain individuals, but the end result will still be the same - finding the real values of x, y, and z that satisfy the equations.