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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 33
Capitolo 6, Problema 33

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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Identify the system of equations: \(x^2 + 4y^2 = 20\) and \(x + 2y = 6\).
From the linear equation \(x + 2y = 6\), solve for one variable in terms of the other. For example, solve for \(x\): \(x = 6 - 2y\).
Substitute the expression for \(x\) into the first equation \(x^2 + 4y^2 = 20\). This gives: \((6 - 2y)^2 + 4y^2 = 20\).
Expand the squared term and simplify the resulting equation to form a quadratic equation in terms of \(y\) only.
Solve the quadratic equation for \(y\), then substitute each \(y\) value back into \(x = 6 - 2y\) to find the corresponding \(x\) values.

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Systems of Equations

A system of equations consists of two or more equations with the same variables. The solution is the set of values that satisfy all equations simultaneously. Understanding how to interpret and solve these systems is fundamental in algebra.
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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.
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Solving Nonlinear Equations

Nonlinear equations, such as those involving squares or other powers, require special techniques like factoring, using the quadratic formula, or isolating terms. Recognizing and solving nonlinear equations is essential when working with systems that include curves or conic sections.
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