From x-y=0 we have y=x . Substituting into 3 x^2-4 y=0 we obtain 3 x^2-4 x=x(3 x-4)=0 . It follo

step 1 So the question says that in problems find all critical points on given plane autonomous system.

From x-y=0 we have y=x . Substituting into 3 x^2-4 y=0 we...

Key Concepts

Zero Product Property

The zero product property states that if a product of two or more factors equals zero, then at least one of the factors must be zero. This property is fundamental in solving factored equations, as it allows one to set each individual factor equal to zero in order to find the solutions for the variable.

Factoring Technique

Factoring is a process of rewriting an expression as a product of its factors. In solving equations, factoring can simplify the process of finding solutions, especially for polynomial equations, by setting each factor equal to zero, which leverages the structure of the equation to break it down into simpler, more manageable parts.

Critical Points

Critical points in a system refer to the values of the variables where the behavior of a function or a dynamical system changes, typically corresponding to stationary points or equilibrium states. They are identified by solving the system of equations derived from setting the derivative, gradient, or similar function to zero, and they often provide insight into the qualitative behavior of the system.

System of Equations

A system of equations involves finding values for the variables that satisfy all the given equations simultaneously. It often requires the use of multiple algebraic methods to reduce, substitute, or manipulate the equations in order to isolate the unknowns and determine the points of intersection between curves or surfaces defined by the equations.

Substitution Method

The substitution method is a technique used to solve systems of equations by expressing one variable in terms of others from one of the equations and substituting that expression into the remaining equation(s). This method reduces the system to a single equation with one unknown, thus simplifying the solution process.

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