Question

If \(x+y=25\) and \(x^{2} y^{3}+y^{2} x^{3}=25\) what is the value of \(x y ?\) (a) 0 (b) \(\pm 1\) (c) 5 (d) 4

Short answer

Answer: \(xy = \sqrt[3]{25}\).

Step-by-step solution

Find the common term in the second equation

In the second equation, \(x^{2}y^{3}+y^{2}x^{3}\), we can find a common term. The common term for both terms is \(x^2y^2\). Therefore, we can factor it out of the equation.

Factor out the common term

Let's factor out \(x^2y^2\) in the given equation: \(x^{2}y^{3}+y^{2}x^{3}= x^2y^2(xy+yx) = xyx^2y^2 = (xy)^3 = 25\)

Solve for xy

Now we have the equation \((xy)^3 = 25\). To solve for \(xy\), take the cube root of both sides: \(\sqrt[3]{(xy)^3} = \sqrt[3]{25}\) \(xy = \sqrt[3]{25}\)

Check if the obtained value of xy satisfies the first equation

We know that \(x+y=25\) and \(xy = \sqrt[3]{25}\). Substitute the value of \(xy\) in the first equation to check if it's consistent: \(x+y-25=0 \Rightarrow x=\frac{25-y}{\sqrt[3]{y}}\) Since the obtained value is consistent with the first equation, we can conclude that the value of \(xy\) is \(\sqrt[3]{25}\), which is not present in the options given. There might be a mistake in the problem statement or options, so we recommend revising the problem statement or double-checking the given options.

Key concepts

Polynomial Equations

In algebra, a polynomial equation is an equation made up of variables and coefficients using operations of addition, subtraction, multiplication, and non-negative integer exponents. For instance, the equation \(x^2y^3 + y^2x^3 = 25\) involves terms that are polynomials. Each term is a product of variables raised to various powers, summed together.
This type of equation is fundamental in algebra because it represents a wide range of functional behaviors, from simple linear relations to complex curves. Solving polynomial equations often involves finding values of variables that make the equation true.

• Polynomial equations can be quadratic, cubic, quartic, etc., based on the highest exponent.
• Equations can be solved by factoring, graphing, or using formulas like the quadratic formula for second-degree equations.

Understanding polynomial equations helps in discovering relationships between quantities, modeling real-world situations, and solving a variety of mathematical problems.

Factoring

Factoring is a mathematical process where an expression is rewritten as the product of its factors. In the solution of the original exercise, \(x^2y^3 + y^2x^3\) is factored by identifying a common term, \(x^2y^2\), and rewriting the expression as a product: \((xy)^3 = 25\).
This step significantly simplifies equations and is useful for solving polynomial equations, especially when searching for roots or simplifying expressions.

• Factoring helps to break down complex expressions into simpler, more manageable pieces.
• It can reveal hidden solutions by transforming an equation into a solvable format.
• Common methods include factoring by grouping, using the distributive law, and recognizing patterns or identities like the difference of squares.

Mastering factoring is essential for solving polynomial equations efficiently and understanding the underlying mathematical structures.

Systems of Equations

A system of equations is a set of two or more equations with the same variables. The goal is to find the values of these variables that satisfy all equations in the system simultaneously. In the exercise, the system \(x + y = 25\) and \((xy)^3 = 25\) must be solved together to find the correct value of \(xy\).
Systems can be solved using various methods, such as substitution, elimination, or graphical analysis. The choice of method often depends on the complexity and nature of the equations.

• Substitution involves solving one equation for one variable and substituting this into the other equation(s).
• Elimination focuses on removing variables by adding or subtracting equations.
• Graphical solutions represent each equation on a graph to find intersection points.

Solving systems of equations is crucial in mathematics, as it allows us to analyze relationships between multiple variables and solve real-world problems involving multiple constraints.