Domanda del libro di testo
In Exercises 16–24, write the partial fraction decomposition of each rational expression.3x/(x - 2)(x^2 + 1)
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Ch. 5 - Systems of Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 6, Problema 21
In Exercises 19–28, solve each system by the addition method.
Guida verificata passo dopo passo1
Identify the system of equations:
\(x^2 - 4y^2 = -7\)
\(3x^2 + y^2 = 31\)
To use the addition method, first align the equations so that adding or subtracting them will eliminate one variable. Notice the variables are \(x^2\) and \(y^2\), so treat \(x^2\) and \(y^2\) as separate terms.
Multiply the second equation by 4 to match the coefficient of \(y^2\) in the first equation:
\(4(3x^2 + y^2) = 4(31)\)
which gives
\(12x^2 + 4y^2 = 124\)
Now add the first equation and the new equation:
\((x^2 - 4y^2) + (12x^2 + 4y^2) = -7 + 124\)
Simplify the left side to eliminate \(y^2\) terms and combine like terms.
Solve the resulting equation for \(x^2\), then substitute this value back into one of the original equations to find \(y^2\). Finally, take square roots to find \(x\) and \(y\) values, remembering to consider both positive and negative roots.
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Systems 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
Addition Method (Elimination Method)
The addition method involves adding or subtracting equations to eliminate one variable, simplifying the system to a single equation with one variable. This technique is useful when equations are arranged so that adding them cancels out a variable, making it easier to solve.
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Handling Quadratic Terms in Systems
When systems include quadratic terms like x² and y², solving requires careful manipulation, such as combining like terms or substitution after elimination. Recognizing the structure of quadratic expressions helps in applying the addition method effectively to find all possible solutions.
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6:19Systems of Inequalities
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