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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 31
Capitolo 6, Problema 31

In Exercises 29–42, solve each system by the method of your choice. {2x2+y2=18xy=4\(\begin{cases}\) 2x^2 + y^2 = 18 \\ xy = 4 \(\end{cases}\)

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Identify the system of equations: \\ \(2x^2 + y^2 = 18\) and $xy = 4$.
From the second equation $xy = 4$, express \(y\) in terms of \(x\(: \\ \)y = \frac{4}{x}\) (assuming \(x \neq 0\)).
Substitute \(y = \frac{4}{x}\) into the first equation \(2x^2 + y^2 = 18\(: \\ \)2x^2 + \left(\frac{4}{x}\right)^2 = 18\).
Simplify the substituted equation: \\ \(2x^2 + \frac{16}{x^2} = 18\).
Multiply both sides of the equation by $x^2\( to clear the denominator and form a polynomial equation: \\ \)2x^4 + 16 = 18x^2$.

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Systems of Equations

A system of equations consists of two or more equations with the same set of variables. The goal is to find values for the variables that satisfy all equations simultaneously. Systems can be solved using various methods such as substitution, elimination, or graphing.
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Nonlinear Equations

Nonlinear equations involve variables raised to powers other than one or multiplied together, such as quadratic terms or products of variables. Solving systems with nonlinear equations often requires special techniques like substitution or factoring, as linear methods may not apply directly.
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Nonlinear Inequalities

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, especially useful when one equation is simpler to isolate a variable.
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In Exercises 29–42, solve each system by the method of your choice. {x2+4y2=20x+2y=6\(\begin{cases}\) x^2 + 4y^2 = 20 \\ x + 2y = 6 \(\end{cases}\)

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In Exercises 31–42, solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. x = 9-2y x + 2y = 13

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Graph the solution set of each system of inequalities or indicate that the system has no solution.

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Write the partial fraction decomposition of each rational expression. 5x2+6x+3/(x + 1)(x² + 2x + 2)

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Write the partial fraction decomposition of each rational expression. x+4/x² (x²+4)

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Graph the solution set of each system of inequalities or indicate that the system has no solution.

{y>2x−3y<−x+6\(\begin{cases}\) y > 2x - 3 \\ y < -x + 6 \(\end{cases}\)

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