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Ch. 5 - Systems of Equations and Inequalities
Blitzer - College Algebra 8th Edition
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Tutti i libri di testoBlitzer 8th EditionCh. 5 - Systems of Equations and InequalitiesProblema 13
Capitolo 6, Problema 13

In Exercises 1–18, solve each system by the substitution method. {xy=3x2+y2=10\(\begin{cases}\) xy = 3 \\ x^2 + y^2 = 10 \(\end{cases}\)

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Start with the given system of equations: $xy = 3$ and \(x^{2} + y^{2} = 10\).
From the first equation $xy = 3$, solve for one variable in terms of the other. For example, solve for \(y\): \(y = \frac{3}{x}\).
Substitute the expression for \(y\) into the second equation \(x^{2} + y^{2} = 10\). This gives: \(x^{2} + \left(\frac{3}{x}\right)^{2} = 10\).
Simplify the substituted equation: \(x^{2} + \frac{9}{x^{2}} = 10\). To clear the denominator, multiply both sides of the equation by \(x^{2}\), resulting in \(x^{4} + 9 = 10x^{2}\).
Rewrite the equation as a quadratic in terms of \(x^{2}\): \(x^{4} - 10x^{2} + 9 = 0\). Let \(u = x^{2}\), then solve the quadratic equation \(u^{2} - 10u + 9 = 0\) for \(u\).

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Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable, making it easier to solve. It is especially useful when one equation is already solved for a variable or can be easily manipulated.
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Solving Nonlinear Systems

Nonlinear systems include equations where variables are raised to powers or multiplied together, such as xy=3 and x² + y²=10. Solving these requires careful algebraic manipulation, often leading to quadratic equations. Understanding how to handle nonlinear terms is essential to find all possible solutions.
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Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically in the form ax² + bx + c = 0. When using substitution, the system often reduces to a quadratic equation, which can be solved using factoring, completing the square, or the quadratic formula. Recognizing and solving quadratics is key to finding the values of variables.
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Introduction to Quadratic Equations
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