In Exercises 1–18, solve each system by the substitution method. xy=3, x^2+y^2=10

Verified step by step guidance

1. Step 1: Solve the first equation for one variable in terms of the other. From the equation xy = 3, express y in terms of x: y = \frac{3}{x}.
2. Step 2: Substitute the expression for y into the second equation. Replace y in the equation x^2 + y^2 = 10 with \frac{3}{x} to get x^2 + \left(\frac{3}{x}\right)^2 = 10.
3. Step 3: Simplify the equation obtained in Step 2. Combine like terms and simplify the expression to form a quadratic equation in terms of x: x^2 + \frac{9}{x^2} = 10.
4. Step 4: Multiply through by x^2 to clear the fraction. This will give you a quadratic equation: x^4 - 10x^2 + 9 = 0.
5. Step 5: Solve the quadratic equation for x. Factorize or use the quadratic formula to find the values of x, and then substitute back to find the corresponding values of y using y = \frac{3}{x}.

Key Concepts

Substitution Method

The substitution method is a technique used to solve systems of equations. It involves solving one equation for one variable and then substituting that expression into the other equation. This method simplifies the system, allowing for easier computation and solution finding.

Systems of Equations

A system of equations consists of two or more equations with the same set of variables. The solution to the system is the set of values that satisfy all equations simultaneously. Understanding how to manipulate and solve these systems is fundamental in algebra, particularly when dealing with linear and nonlinear equations.

Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically in the form ax^2 + bx + c = 0. In the context of the given problem, the equation x^2 + y^2 = 10 represents a circle, which is a nonlinear relationship. Recognizing the nature of these equations is crucial for applying the substitution method effectively.