Find the Domain -7y^2+zy-x=0

Problem

−7*y2+z*y−x=0

Solution

1. Identify the goal, which is to find the domain of the relation. In a multivariable context where the equation is not explicitly solved for one variable, the domain typically refers to the set of all possible values for the independent variables (x,z) such that a real value for the dependent variable y exists.

2. Rearrange the equation into the standard form of a quadratic equation in terms of y

−7*y2+z*y−x=0

3. Apply the quadratic formula to solve for y in terms of x and z For an equation a*y2+b*y+c=0 the solution is y=(−b±√(,b2−4*a*c))/(2*a)

y=(−z±√(,z2−4*(−7)*(−x)))/(2*(−7))

4. Simplify the expression inside the square root (the discriminant).

y=(−z±√(,z2−28*x))/(−14)

5. Determine the condition for y to be a real number. The expression inside the square root must be greater than or equal to zero.

z2−28*x≥0

6. Solve the inequality to define the domain in terms of x and z

z2≥28*x

x≤(z2)/28

Final Answer

Domain={[(x,z)∈ℝ2,x≤(z2)/28]}

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