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Ch. 5 - Systems and Matrices
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 5 - Systems and MatricesProblem 34
Chapter 6, Problem 34

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.
5x2−2y2=255x^2 - 2y^2 = 25
10x2+y2=5010x^2 + y^2 = 50

Verified step by step guidance
1
Start by writing down the system of equations clearly: \(5x^2 - 2y^2 = 25\) \(10x^2 + y^2 = 50\)
To solve the system, consider using substitution or elimination. Notice both equations involve \(x^2\) and \(y^2\), so treat \(x^2\) and \(y^2\) as variables to simplify the process.
Multiply the first equation by a suitable number so that the coefficients of either \(x^2\) or \(y^2\) match in both equations. For example, multiply the first equation by 1 to keep it as is, and the second equation by 2 to align the \(y^2\) terms:
\(5x^2 - 2y^2 = 25\) \(20x^2 + 2y^2 = 100\)
Add the two equations to eliminate \(y^2\): \((5x^2 - 2y^2) + (20x^2 + 2y^2) = 25 + 100\) Simplify to find an equation in terms of \(x^2\) only, then solve for \(x^2\). After finding \(x^2\), substitute back into one of the original equations to solve for \(y^2\). Finally, take square roots to find \(x\) and \(y\), remembering to consider both positive and negative roots as well as complex solutions.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Nonlinear Systems of Equations

A nonlinear system consists of two or more equations involving variables raised to powers other than one or multiplied together. Solving such systems requires methods beyond simple substitution or elimination used for linear systems, often involving algebraic manipulation or substitution to reduce the system to solvable equations.
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Substitution and Elimination Methods

These are algebraic techniques used to solve systems of equations. Substitution involves solving one equation for a variable and substituting into the other, while elimination involves adding or subtracting equations to eliminate a variable. Both methods help reduce the system to a single equation in one variable.
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Complex Solutions in Algebra

When solving equations, solutions may include nonreal complex numbers, especially if the equation involves squares or other powers that can yield negative values under radicals. Recognizing and including complex solutions ensures a complete solution set, which is important in college algebra.
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Related Practice
Textbook Question

Solve each system of equations. State whether it is an inconsistent system or has infinitely many solutions. If a system has infinitely many solutions, write the solution set with x arbitrary.

5x - 5y - 3 = 0

x - y - 12 = 0

Textbook Question

Find each sum or difference, if possible.

[−43]−[582]\(\left\)[ \(\begin{matrix}\) -4 & 3 \(\end{matrix}\) \(\right\)] - \(\left\)[ \(\begin{matrix}\) 5 & 8 & 2 \(\end{matrix}\) \(\right\)]

Textbook Question

Find each sum or difference, if possible. See Examples 2 and 3.

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Textbook Question

Match each inequality with the appropriate calculator graph in A–D. Do not use a calculator y ≤ -3x - 6

Textbook Question

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (-x4 - 8x2 + 3x - 10)/((x + 2)(x2 + 4)2)

Textbook Question

Work each problem. Write the inequality that represents the region inside a circle with center (-5, -2) and radius 4.