Question

Solve each system. Use any method you wish. $$ \left\\{\begin{array}{r} x^{3}-2 x^{2}+y^{2}+3 y-4=0 \\ x-2+\frac{y^{2}-y}{x^{2}}=0 \end{array}\right. $$

Short answer

The solution is \( (x, y) = (2, 1) \).

Step-by-step solution

Analyze the system of equations

The given system of equations is: \[\begin{cases} x^{3}-2x^{2}+y^{2}+3y-4=0 \ x-2+\frac{y^{2}-y}{x^{2}}=0\end{cases}\]The first equation is a polynomial in terms of both \(x\) and \(y\). The second equation can be simplified for easier handling.

Simplify the second equation

Rewrite the second equation:\[ x = 2 - \frac{y^{2} - y}{x^{2}} \]Multiply both sides by \(x^2\) to clear the fraction:\[ x^3 = 2x^2 - (y^2 - y) \]

Substitute for \(x^3\) in the first equation

Replace \(x^3\) in the first equation with the expression obtained from the second equation:\[2x^2 - (y^2 - y) - 2x^2 + y^2 + 3y - 4 = 0 \]Simplify the equation:\[- y^2 + y + y^2 + 3y - 4 = 0 \]Combine like terms:\[4y - 4 = 0 \]

Solve for \(y\)

Divide both sides by 4:\[ y - 1 = 0 \]Thus, \( y = 1 \).

Solve for \(x\)

Substitute \( y = 1 \) back into the simplified second equation to find \(x\):\[x = 2 - \frac{1^2 - 1}{x^2} = 2 \]Since there are no contradictions, the solution is valid.

Key concepts

Polynomial Equations

Polynomial equations are mathematical expressions involving sums of powers of variables where the exponents are non-negative integers. In the given system, the first equation is a polynomial equation: \( x^{3} - 2x^{2} + y^{2} + 3y - 4 = 0 \). This specific equation is a combination of terms with different degrees in variables x and y. Such equations are crucial because they help us understand relationships between variables and solve for unknowns.

When dealing with polynomial equations, it's often useful to become familiar with terms like:

• Degree: The highest power of the variable in the polynomial. Here, the degree is 3 due to the term \( x^{3} \).
• Coefficient: Numbers that multiply each term in the polynomial, like -2 with \( x^{2} \).
• Constant: A term without any variables, such as -4 in the equation.

Understanding these terms and simplifying polynomial equations as much as possible can make solving them much easier.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into another equation. In the provided system, we simplify the second equation for x: \[ x = 2 - \frac{y^{2} - y}{x^{2}} \].

Follow these steps:

• Isolate one variable in one of the equations.
• Substitute the isolated variable's expression into the other equation.
• Simplify the resulting equation and solve for the variable.

In this case, multiplying both sides by \( x^{2} \) cleared the fraction, resulting in a polynomial in terms of x and y that matched our first equation's format. This substitution allowed us to combine and simplify the equations effectively.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a key skill in solving systems of equations. It involves operations that make the expression easier to handle. For our second equation, we simplified: \[ x = 2 - \frac{y^{2} - y}{x^{2}} \].

Steps to simplify an algebraic expression typically include:

• Clearing any fractions by multiplying by common denominators.
• Combining like terms to gather similar variable terms.
• Isolating terms involving the same variable together.

In our solution, we multiplied by \( x^{2} \) to simplify, then substituted this simplified form into the first equation to reduce complexity. After combining like terms, we concluded that \( 4y - 4 = 0 \), leading us directly to \( y = 1 \). Then substituting \( y = 1 \) back helped us solve for \( x = 2 \). Simplification plays a pivotal role in reducing the problem to simpler forms that are easier to solve.