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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 29
Capitolo 6, Problema 29

In Exercises 29–42, solve each system by the method of your choice. {3x2+4y2=162x2−3y2=5\(\begin{cases}\) 3x^2 + 4y^2 = 16 \\ 2x^2 - 3y^2 = 5 \(\end{cases}\)

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1
Identify the system of equations to solve: \[3x^2 + 4y^2 = 16\] \[2x^2 - 3y^2 = 5\]
To solve the system, notice both equations involve \(x^2\) and \(y^2\). Let’s introduce new variables to simplify: Let \[a = x^2\] and \[b = y^2\]. Then rewrite the system as: \[3a + 4b = 16\] \[2a - 3b = 5\]
Solve the system of linear equations in terms of \(a\) and \(b\). For example, use the method of substitution or elimination: - Multiply one or both equations to align coefficients, - Then add or subtract the equations to eliminate one variable, - Solve for the remaining variable.
Once you find the values of \(a\) and \(b\), recall that \(a = x^2\) and \(b = y^2\). To find \(x\) and \(y\), take the square root of each value: \[x = \pm \sqrt{a}\] \[y = \pm \sqrt{b}\]
Check each pair \((x, y)\) in the original equations to verify the solutions, since squaring can introduce extraneous solutions.

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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 set up systems is essential before applying any solving method.
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Solving Systems by Substitution or Elimination

These are common methods to solve systems. Substitution involves solving one equation for a variable and substituting into the other. Elimination involves adding or subtracting equations to eliminate a variable. Both methods can be adapted for nonlinear systems like the given quadratic equations.
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Handling Quadratic Equations in Systems

When systems include quadratic terms, solutions may be multiple or complex. Recognizing how to manipulate and combine quadratic expressions, such as factoring or isolating terms, is crucial. This helps in reducing the system to simpler forms or single-variable equations.
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