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Ch. 5 - Systems of Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 5 - Systems of Equations and InequalitiesProblema 9
Capitolo 6, Problema 9

In Exercises 1–18, solve each system by the substitution method. {xy=62x−y=1\(\begin{cases}\) xy = 6 \\ 2x - y = 1 \(\end{cases}\)

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Start with the given system of equations: $xy = 6$ and \(2x - y = 1\).
From the second equation \(2x - y = 1\), solve for \(y\) in terms of \(x\): \(y = 2x - 1\).
Substitute the expression for \(y\) into the first equation $xy = 6$: \(x(2x - 1) = 6\).
Expand and rewrite the equation: \(2x^2 - x = 6\).
Rearrange the equation to standard quadratic form: \(2x^2 - x - 6 = 0\), then solve for \(x\) using the quadratic formula or factoring.

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System of Equations

A system of equations consists of two or more equations with the same variables. The goal is to find values for the variables that satisfy all equations simultaneously. Understanding how to interpret and manipulate these systems is essential for finding their solutions.
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Introduction to Systems of Linear Equations

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 easily solved for one variable.
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Choosing a Method to Solve Quadratics

Solving Nonlinear Equations

When a system includes nonlinear equations, such as xy = 6, solving requires careful algebraic manipulation. After substitution, the resulting equation may be quadratic or another nonlinear form, requiring techniques like factoring or the quadratic formula to find solutions.
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Nonlinear Inequalities
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